1,561 results
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2. A Look at Robustness and Stability of $\ell_{1}$-versus $\ell_{0}$-Regularization: Discussion of Papers by Bertsimas et al. and Hastie et al
- Author
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Peter Bühlmann, Armeen Taeb, and Yuansi Chen
- Subjects
Statistics and Probability ,latent variables ,low-rank estimation ,General Mathematics ,Linear model ,020206 networking & telecommunications ,Feature selection ,02 engineering and technology ,Latent variable ,01 natural sciences ,Regularization (mathematics) ,010104 statistics & probability ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,Distributional robustness ,0101 mathematics ,Statistics, Probability and Uncertainty ,high-dimensional estimation ,Mathematics ,variable selection - Abstract
We congratulate the authors Bertsimas, Pauphilet and van Parys (hereafter BPvP) and Hastie, Tibshirani and Tibshirani (hereafter HTT) for providing fresh and insightful views on the problem of variable selection and prediction in linear models. Their contributions at the fundamental level provide guidance for more complex models and procedures.
- Published
- 2020
- Full Text
- View/download PDF
3. From coin tossing to rock-paper-scissors and beyond: a log-exp gap theorem for selecting a leader
- Author
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Hsien-Kuei Hwang, Yoshiaki Itoh, and Michael Fuchs
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Statistics and Probability ,Discrete mathematics ,Class (set theory) ,Coin flipping ,Group (mathematics) ,General Mathematics ,Variance (accounting) ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Logarithmic mean ,Bounded function ,0103 physical sciences ,Gap theorem ,0101 mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,Mathematics - Abstract
A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.
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- 2017
4. A Note on a Paper by Wong and Heyde
- Author
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Mikhail Urusov and Aleksandar Mijatović
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Statistics and Probability ,Statistics::Theory ,Pure mathematics ,60G44, 60G48, 60H10, 60J60 ,General Mathematics ,Applied probability ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,60G48 ,FOS: Mathematics ,60G44 ,0101 mathematics ,60J60 ,Mathematics ,Local martingales versus true martingales ,010102 general mathematics ,Probability (math.PR) ,stochastic exponential ,Exponential function ,Mathematik ,60H10 ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Quantitative Finance - General Finance ,General Finance (q-fin.GN) ,Mathematics - Probability ,Counterexample - Abstract
In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed., Comment: To appear in Journal of Applied Probability, 11 pages
- Published
- 2011
5. Epidemics with carriers: A note on a paper of Dietz
- Author
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F. Downton
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Statistics and Probability ,Entire population ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,Short interval ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Demography ,Mathematics - Abstract
In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.
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- 1967
6. d-Hermite rings and skew $$\textit{PBW}$$ PBW extensions
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Oswaldo Lezama and Claudia Gallego
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Hermite polynomials ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Short paper ,Skew ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,symbols.namesake ,Computational Theory and Mathematics ,Kronecker delta ,symbols ,Kronecker's theorem ,Finitely-generated abelian group ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this short paper we study the d-Hermite condition about stably free modules for skew $$\textit{PBW}$$ extensions. For this purpose, we estimate the stable rank of these non-commutative rings. In addition, and closely related with these questions, we will prove Kronecker’s theorem about the radical of finitely generated ideals for some particular types of skew $$\textit{PBW}$$ extensions.
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- 2015
7. Integral Representations for the Hartman–Watson Density
- Author
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Yuu Hariya
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Statistics and Probability ,Statistics::Theory ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Expression (computer science) ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Mathematics::Probability ,Simple (abstract algebra) ,FOS: Mathematics ,60J65 (Primary), 60J55, 60E10 (Secondary) ,Integral formula ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Brownian motion ,Mathematics ,Mathematical physics - Abstract
This paper concerns the density of the Hartman--Watson law. Yor (1980) obtained an integral formula that gives a closed-form expression of the Hartman--Watson density. In this paper, based on Yor's formula, we provide alternative integral representations for the density. As an immediate application, we recover in part a Dufresne's result (2001) that exhibits remarkably simple representations for the laws of exponential additive functionals of Brownian motion., Comment: 21 pages; this is an abridged version of the previous one, accepted for publication in the Journal of Theoretical Probability, with shortened Section 4
- Published
- 2021
8. On a new stochastic model for cascading failures
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Hyunju Lee
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Statistics and Probability ,Stochastic modelling ,General Mathematics ,010102 general mathematics ,Residual ,01 natural sciences ,Stochastic ordering ,Cascading failure ,010104 statistics & probability ,Control theory ,Component (UML) ,Life test ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.
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- 2020
9. On the classification of simple Lie algebras of dimension seven over fields of characteristic 2
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Wilian Francisco de Araujo, Marinês Guerreiro, and Alexander Grishkov
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Pure mathematics ,Rank (linear algebra) ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,SUPERÁLGEBRAS DE LIE ,Field (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Computational Theory and Mathematics ,Simple (abstract algebra) ,0103 physical sciences ,Lie algebra ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algebra over a field ,Algebraically closed field ,Mathematics - Abstract
This paper is the second part of paper (Grishkov and Guerreiro in Sao Paulo J Math Sci v4(1):93–107, 2010) about simple 7-dimensional Lie algebras over an algebraically closed field k of characteristic two. In this paper we prove that all simple 7-dimensional Lie algebras over k of absolute toral rank three are isomorphic to the Cartan algebra $$W_1$$ or the Hamilton algebra $$H_2.$$ We hope to prove that those algebras are the unique simple 7-dimensional Lie algebras over the field k. Observe that in the case of absolute toral rank 2 this fact was proved in [2].
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- 2020
10. Explicit descriptions of spectral properties of Laplacians on spheres $${\mathbb {S}}^{N}\,(N\ge 1)$$: a review
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Richard Olu Awonusika
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Pure mathematics ,Parametrix ,General Mathematics ,010102 general mathematics ,Spectral geometry ,0102 computer and information sciences ,Mathematics::Spectral Theory ,Riemannian manifold ,01 natural sciences ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Heat equation ,0101 mathematics ,Statistics, Probability and Uncertainty ,Asymptotic expansion ,Laplace operator ,Eigenvalues and eigenvectors ,Heat kernel ,Mathematics - Abstract
In their remarkable paper, Minakshisundaram and Pleijel established by using the parametrix for the heat equation the asymptotic expansion of the heat kernel on compact Riemannian manifolds. The result has since been extensively used in the spectral analysis of the Laplace-Beltrami operator, and in particular, in proving Weyl’s law for the asymptotic distribution of eigenvalues and various direct and inverse problems in spectral geometry. However, the question of describing the explicit values of the corresponding heat trace coefficients associated with an arbitrary compact Riemannian manifold has remained an interesting task. In this paper, we review results on Minakshisundaram-Pleijel coefficients associated with the Laplacian on spheres $${\mathbb {S}}^{N}$$ ( $$N\ge 1$$ ) and other associated spectral invariants, namely, the Minakshisundaram-Pleijel zeta functions & their residues, and the zeta-regularised determinants of the Laplacian on spheres. The results reviewed deal mainly with closed-form formulae for the afore-mentioned spectral invariants and the explicit values of the first few of these spectral invariants are given.
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- 2020
11. On moderate deviations in Poisson approximation
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Qingwei Liu and Aihua Xia
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Statistics and Probability ,Random graph ,Matching (graph theory) ,Distribution (number theory) ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,Birthday problem ,Normal distribution ,010104 statistics & probability ,symbols.namesake ,FOS: Mathematics ,Rare events ,symbols ,Applied mathematics ,Moderate deviations ,0101 mathematics ,Statistics, Probability and Uncertainty ,Primary 60F05, secondary 60E15 ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures
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- 2020
12. Theorems connecting Stieltjes transform and Hankel transform
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Virendra Kumar
- Subjects
Pure mathematics ,Hankel transform ,Computational Theory and Mathematics ,Special functions ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Stieltjes transform ,Mathematics - Abstract
The aim of the present paper is to establish four theorems connecting Stieltjes transform and Hankel transform. By applications of the theorems established in this paper four new integral formulae involving special functions are obtained. Due to the general nature of the theorems established in this paper, several other integrals involving special functions may be evaluated.
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- 2020
13. Time-Changed Local Martingales Under Signed Measures
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Sakrani Samia
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Statistics and Probability ,Discrete mathematics ,General Mathematics ,Signed measure ,010102 general mathematics ,Stochastic calculus ,01 natural sciences ,Time changes ,Stochastic integration ,010104 statistics & probability ,Mathematics::Probability ,Bounded function ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Brownian motion ,Mathematics - Abstract
In this paper, we use stochastic integration in the framework of signed measures, together with the technique of time changes. Let Q be a bounded non-null signed measure on $$(\varOmega ,{\mathcal {F}}, {{P}}),$$ such that $$\left| Q\right| $$ and P are equivalent. In the first part of the paper, we generalize the results of stochastic calculus in Beghdadi-Sakrani (Seminaire de probabilites XXXVI, Springer, 2003) to Q-local martingales and we give some examples. In the second part, we prove that the class of Q-semimartingales is invariant under time changes. We establish the famous formulas of time-changed local martingales as well as the representation of a Q-local martingale as a time-changed Brownian motion.
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- 2020
14. Martingale decomposition of an L2 space with nonlinear stochastic integrals
- Author
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Clarence Simard
- Subjects
Statistics and Probability ,Optimization problem ,General Mathematics ,010102 general mathematics ,Stochastic calculus ,01 natural sciences ,010104 statistics & probability ,Nonlinear system ,Integrator ,Bounded function ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Lp space ,Martingale (probability theory) ,Brownian motion ,Mathematics - Abstract
This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.
- Published
- 2019
15. On the Convergence of FK–Ising Percolation to $$\mathrm {SLE}(16/3, (16/3)-6)$$
- Author
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Christophe Garban and Hao Wu
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Statistics and Probability ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,Order (ring theory) ,01 natural sciences ,010104 statistics & probability ,Scaling limit ,Mathematics::Probability ,Conformal symmetry ,Chordal graph ,Exponent ,Ising model ,Continuum (set theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK–Ising percolation to chordal $$\mathrm {SLE}_\kappa ( \kappa -6)$$ with $$\kappa =16/3$$. Our proof follows the classical excursion construction of $$\mathrm {SLE}_\kappa (\kappa -6)$$ processes in the continuum, and we are thus led to introduce suitable cut-off stopping times in order to analyse the behaviour of the driving function of the discrete system when Dobrushin boundary condition collapses to a single point. Our proof is very different from that of Kemppainen and Smirnov (Conformal invariance of boundary touching loops of FK–Ising model. arXiv:1509.08858, 2015; Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE. arXiv:1609.08527, 2016) as it only relies on the convergence to the chordal $$\mathrm {SLE}_{\kappa }$$ process in Dobrushin boundary condition and does not require the introduction of a new observable. Still, it relies crucially on several ingredients: One important emphasis of this paper is to carefully write down some properties which are often considered folklore in the literature but which are only justified so far by hand-waving arguments. The main examples of these are: We end the paper with a detailed sketch of the convergence to radial $$\mathrm {SLE}_\kappa ( \kappa -6)$$ when $$\kappa =16/3$$ as well as the derivation of Onsager’s one-arm exponent 1 / 8.
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- 2019
16. Resolvent Decomposition Theorems and Their Application in Denumerable Markov Processes with Instantaneous States
- Author
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Anyue Chen
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Probabilistic logic ,Markov process ,01 natural sciences ,Interpretation (model theory) ,Algebra ,010104 statistics & probability ,symbols.namesake ,symbols ,Decomposition (computer science) ,Countable set ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Resolvent ,Mathematics ,Analytic proof - Abstract
The basic aim of this paper is to provide a fundamental tool, the resolvent decomposition theorem, in the construction theory of denumerable Markov processes. We present a detailed analytic proof of this extremely useful tool and explain its clear probabilistic interpretation. We then apply this tool to investigate the basic problems of existence and uniqueness criteria for denumerable Markov processes with instantaneous states to which few results have been obtained even until now. Although the complete answers regarding these existence and uniqueness criteria will be given in a subsequent paper, we shall, in this paper, present part solutions of these very important problems that are closely linked with the subtle Williams S and N conditions.
- Published
- 2019
17. Approximate lumpability for Markovian agent-based models using local symmetries
- Author
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Wasiur R. KhudaBukhsh, Arnab Auddy, Heinz Koeppl, and Yann Disser
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Statistics and Probability ,Random graph ,Markov chain ,General Mathematics ,Probability (math.PR) ,Lumpability ,Neighbourhood (graph theory) ,Markov process ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,60J28 ,010201 computation theory & mathematics ,Approximation error ,Local symmetry ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,State space ,Applied mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Mathematics - Abstract
We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed., Comment: 28 pages, 4 figures
- Published
- 2019
18. Comparison results for M/G/1 queues with waiting and sojourn time deadlines
- Author
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Yoshiaki Inoue
- Subjects
Statistics and Probability ,Waiting time ,Discrete mathematics ,021103 operations research ,Service time ,General Mathematics ,0211 other engineering and technologies ,Comparison results ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,M/G/1 queue ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.
- Published
- 2019
19. Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I
- Author
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Robert E. Gaunt
- Subjects
Statistics and Probability ,Mathematics(all) ,General Mathematics ,Stein’s method ,Computer Science::Digital Libraries ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Variance-gamma approximation ,Applied mathematics ,0101 mathematics ,Gaussian process ,Mathematics ,Laplace transform ,010102 general mathematics ,Distributional transformation ,Stein's method ,Rate of convergence ,Variance-gamma distribution ,Distribution (mathematics) ,Laplace's method ,symbols ,Statistics, Probability and Uncertainty ,Random variable - Abstract
The variance-gamma (VG) distributions form a four-parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein’s method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper (Gaunt 2018), which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.
- Published
- 2018
20. Optimal stopping for the exponential of a Brownian bridge
- Author
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Tiziano De Angelis and Alessandro Milazzo
- Subjects
Statistics and Probability ,General Mathematics ,Structure (category theory) ,Expected value ,Bond/stock selling ,Free boundary problems ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,Mathematics::Probability ,Bellman equation ,Optimal stopping ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,Mathematics ,Brownian bridge ,Continuous boundary ,Regularity of value function ,010102 general mathematics ,Probability (math.PR) ,Mathematical Finance (q-fin.MF) ,Exponential function ,Quantitative Finance - Mathematical Finance ,Optimization and Control (math.OC) ,Optimal stopping rule ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Mathematics - Probability - Abstract
In this paper we study the problem of stopping a Brownian bridge $X$ in order to maximise the expected value of an exponential gain function. In particular, we solve the stopping problem $$\sup_{0\le \tau\le 1}\mathsf{E}[\mathrm{e}^{X_\tau}]$$ which was posed by Ernst and Shepp in their paper [Commun. Stoch. Anal., 9 (3), 2015, pp. 419--423] and was motivated by bond selling with non-negative prices. Due to the non-linear structure of the exponential gain, we cannot rely on methods used in the literature to find closed-form solutions to other problems involving the Brownian bridge. Instead, we develop techniques that use pathwise properties of the Brownian bridge and martingale methods of optimal stopping theory in order to find the optimal stopping rule and to show regularity of the value function., Comment: 22 pages, 6 figures
- Published
- 2020
21. Maximum Independent Component Analysis with Application to EEG Data
- Author
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Zhengjun Zhang, Chunming Zhang, and Ruosi Guo
- Subjects
Statistics and Probability ,Discrete mathematics ,0303 health sciences ,nonlinear time series ,General Mathematics ,Crossover ,Contrast (statistics) ,01 natural sciences ,Independent component analysis ,Blind signal separation ,Linear map ,010104 statistics & probability ,03 medical and health sciences ,Nonlinear system ,image analysis ,Genetic algorithm ,genetic algorithm ,Blind source separation ,0101 mathematics ,Statistics, Probability and Uncertainty ,Linear combination ,optimization ,030304 developmental biology ,Mathematics - Abstract
In many scientific disciplines, finding hidden influential factors behind observational data is essential but challenging. The majority of existing approaches, such as the independent component analysis (${\mathrm{ICA}}$), rely on linear transformation, that is, true signals are linear combinations of hidden components. Motivated from analyzing nonlinear temporal signals in neuroscience, genetics, and finance, this paper proposes the “maximum independent component analysis” (${\mathrm{MaxICA}}$), based on max-linear combinations of components. In contrast to existing methods, ${\mathrm{MaxICA}}$ benefits from focusing on significant major components while filtering out ignorable components. A major tool for parameter learning of ${\mathrm{MaxICA}}$ is an augmented genetic algorithm, consisting of three schemes for the elite weighted sum selection, randomly combined crossover, and dynamic mutation. Extensive empirical evaluations demonstrate the effectiveness of ${\mathrm{MaxICA}}$ in either extracting max-linearly combined essential sources in many applications or supplying a better approximation for nonlinearly combined source signals, such as $\mathrm{EEG}$ recordings analyzed in this paper.
- Published
- 2020
22. A multiplicatively symmetrized version of the Chung-Diaconis-Graham random process
- Author
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Martin Hildebrand
- Subjects
Statistics and Probability ,Stochastic process ,General Mathematics ,010102 general mathematics ,Probability (math.PR) ,Order (ring theory) ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,60B15 (Primary) 60G50 (Secondary) ,Mathematics - Probability ,Mathematics - Abstract
This paper considers random processes of the form $$X_{n+1}=a_nX_n+b_n\pmod p$$ where p is odd, $$X_0=0$$ , $$(a_0,b_0), (a_1,b_1), (a_2,b_2),\ldots $$ are i.i.d., and $$a_n$$ and $$b_n$$ are independent with $$P(a_n=2)=P(a_n=(p+1)/2)=1/2$$ and $$P(b_n=1)=P(b_n=0)=P(b_n=-1)=1/3$$ . This can be viewed as a multiplicatively symmetrized version of a random process of Chung, Diaconis, and Graham. This paper shows that order $$(\log p)^2$$ steps suffice for $$X_n$$ to be close to uniformly distributed on the integers mod p for all odd p while order $$(\log p)^2$$ steps are necessary for $$X_n$$ to be close to uniformly distributed on the integers mod p.
- Published
- 2020
- Full Text
- View/download PDF
23. Discrete-type approximations for non-Markovian optimal stopping problems: Part I
- Author
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Dorival Leão, Francesco Russo, Alberto Ohashi, Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris), Optimisation et commande (OC), and École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)
- Subjects
Statistics and Probability ,General Mathematics ,Monte Carlo method ,Computational Finance (q-fin.CP) ,01 natural sciences ,FOS: Economics and business ,010104 statistics & probability ,Stochastic differential equation ,Quantitative Finance - Computational Finance ,0502 economics and business ,Filtration (mathematics) ,FOS: Mathematics ,Applied mathematics ,Optimal stopping ,0101 mathematics ,Brownian motion ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Stochastic control ,050208 finance ,Fractional Brownian motion ,05 social sciences ,Probability (math.PR) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Variational inequality ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
In this paper, we present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\epsilon$-optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent SDEs driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte-Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra, Ohashi and Russo., Comment: Final version to appear in Journal of Applied Probability
- Published
- 2019
24. Minimal dispersion approximately balancing weights: asymptotic properties and practical considerations
- Author
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José R. Zubizarreta and Yixin Wang
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Mean squared error ,General Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Statistics - Applications ,01 natural sciences ,Methodology (stat.ME) ,010104 statistics & probability ,Covariate ,FOS: Mathematics ,050602 political science & public administration ,Applied mathematics ,Applications (stat.AP) ,Statistical dispersion ,0101 mathematics ,Statistics - Methodology ,Mathematics ,Smoothness (probability theory) ,Applied Mathematics ,05 social sciences ,Estimator ,Function (mathematics) ,Agricultural and Biological Sciences (miscellaneous) ,0506 political science ,Weighting ,Inverse probability ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences - Abstract
Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods which find the weights of minimum dispersion that approximately balance the covariates. We call these weights "minimal weights" and study them under a common optimization framework. The key observation is the connection between approximate covariate balance and shrinkage estimation of the propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. Also, we show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. From a practical standpoint, we present a finite sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. We finally provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. We conclude the paper with four empirical studies that suggest approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. In these studies, we show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance., 41 pages
- Published
- 2019
25. Central Limit Theorem for Lipschitz–Killing Curvatures of Excursion Sets of Gaussian Random Fields
- Author
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Sreekar Vadlamani and Marie Kratz
- Subjects
Statistics and Probability ,Pure mathematics ,Random field ,Series (mathematics) ,Stochastic process ,General Mathematics ,Gaussian ,010102 general mathematics ,Mathematical analysis ,Excursion ,Lipschitz continuity ,01 natural sciences ,Gaussian random field ,010104 statistics & probability ,symbols.namesake ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Central limit theorem - Abstract
Our interest in this paper is to explore limit theorems for various geometric functionals of excursion sets of isotropic Gaussian random fields. In the past, asymptotics of nonlinear functionals of Gaussian random fields have been studied [see Berman (Sojourns and extremes of stochastic processes, Wadsworth & Brooks, Monterey, 1991), Kratz and Leon (Extremes 3(1):57–86, 2000), Kratz and Leon (J Theor Probab 14(3):639–672, 2001), Meshenmoser and Shashkin (Stat Probab Lett 81(6):642–646, 2011), Pham (Stoch Proc Appl 123(6):2158–2174, 2013), Spodarev (Chapter in modern stochastics and applications, volume 90 of the series Springer optimization and its applications, pp 221–241, 2013) for a sample of works in such settings], the most recent addition being (Adler and Naitzat in Stoch Proc Appl 2016; Estrade and Leon in Ann Probab 2016) where a central limit theorem (CLT) for Euler integral and Euler–Poincare characteristic, respectively, of the excursions set of a Gaussian random field is proven under some conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz–Killing curvatures of excursion sets of Gaussian random fields, in an appropriate setting.
- Published
- 2017
26. Rejoinder: probabilistic integration: a role in statistical computation?
- Author
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Dino Sejdinovic, François-Xavier Briol, Michael A. Osborne, Chris J. Oates, and Mark Girolami
- Subjects
Statistics and Probability ,FOS: Computer and information sciences ,Computer Science - Machine Learning ,math.NA ,Computer science ,probabilistic numerics ,uncertainty quantification ,Statistics & Probability ,General Mathematics ,cs.LG ,Bayesian probability ,Machine Learning (stat.ML) ,01 natural sciences ,Constructive ,Statistics - Computation ,Machine Learning (cs.LG) ,010104 statistics & probability ,03 medical and health sciences ,Statistics - Machine Learning ,Computational statistics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Uncertainty quantification ,Computation (stat.CO) ,030304 developmental biology ,stat.CO ,0303 health sciences ,Science & Technology ,0104 Statistics ,Probabilistic logic ,Nonparametric statistics ,Statistical computation ,Numerical Analysis (math.NA) ,stat.ML ,nonparametric statistics ,Physical Sciences ,Statistics, Probability and Uncertainty ,Mathematical economics ,Mathematics - Abstract
This article is the rejoinder for the paper "Probabilistic Integration: A Role in Statistical Computation?" to appear in Statistical Science with discussion. We would first like to thank the reviewers and many of our colleagues who helped shape this paper, the editor for selecting our paper for discussion, and of course all of the discussants for their thoughtful, insightful and constructive comments. In this rejoinder, we respond to some of the points raised by the discussants and comment further on the fundamental questions underlying the paper: (i) Should Bayesian ideas be used in numerical analysis?, and (ii) If so, what role should such approaches have in statistical computation?, Comment: Accepted to Statistical Science
- Published
- 2019
27. Bootstrap of residual processes in regression: to smooth or not to smooth?
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I Van Keilegom, Natalie Neumeyer, and UCL - SSH/LIDAM/ISBA - Institut de Statistique, Biostatistique et Sciences Actuarielles
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Independent and identically distributed random variables ,Statistics::Theory ,General Mathematics ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,Residual ,01 natural sciences ,Methodology (stat.ME) ,010104 statistics & probability ,0502 economics and business ,FOS: Mathematics ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Statistics - Methodology ,050205 econometrics ,Mathematics ,Applied Mathematics ,05 social sciences ,Nonparametric statistics ,Estimator ,Regression analysis ,Agricultural and Biological Sciences (miscellaneous) ,Empirical distribution function ,Nonparametric regression ,Kernel smoother ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences - Abstract
In this paper we consider a location model of the form $Y = m(X) + \varepsilon$, where $m(\cdot)$ is the unknown regression function, the error $\varepsilon$ is independent of the $p$-dimensional covariate $X$ and $E(\varepsilon)=0$. Given i.i.d. data $(X_1,Y_1),\ldots,(X_n,Y_n)$ and given an estimator $\hat m(\cdot)$ of the function $m(\cdot)$ (which can be parametric or nonparametric of nature), we estimate the distribution of the error term $\varepsilon$ by the empirical distribution of the residuals $Y_i-\hat m(X_i)$, $i=1,\ldots,n$. To approximate the distribution of this estimator, Koul and Lahiri (1994) and Neumeyer (2008, 2009) proposed bootstrap procedures, based on smoothing the residuals either before or after drawing bootstrap samples. So far it has been an open question whether a classical non-smooth residual bootstrap is asymptotically valid in this context. In this paper we solve this open problem, and show that the non-smooth residual bootstrap is consistent. We illustrate this theoretical result by means of simulations, that show the accuracy of this bootstrap procedure for various models, testing procedures and sample sizes.
- Published
- 2019
- Full Text
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28. Monte Carlo Fusion
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Gareth O. Roberts, Murray Pollock, and Hongsheng Dai
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Theoretical computer science ,Unification ,business.industry ,General Mathematics ,Big data ,Monte Carlo method ,Inference ,Expert elicitation ,02 engineering and technology ,Construct (python library) ,01 natural sciences ,Methodology (stat.ME) ,010104 statistics & probability ,Approximation error ,65C05, 65C60, 62C10, 65C30 ,0202 electrical engineering, electronic engineering, information engineering ,Key (cryptography) ,020201 artificial intelligence & image processing ,0101 mathematics ,Statistics, Probability and Uncertainty ,business ,Statistics - Methodology ,Mathematics - Abstract
In this paper we propose a new theory and methodology to tackle the problem of unifying Monte Carlo samples from distributed densities into a single Monte Carlo draw from the target density. This surprisingly challenging problem arises in many settings (for instance, expert elicitation, multiview learning, distributed ‘big data’ problems, etc.), but to date the framework and methodology proposed in this paper (Monte Carlo fusion) is the first general approach which avoids any form of approximation error in obtaining the unified inference. In this paper we focus on the key theoretical underpinnings of this new methodology, and simple (direct) Monte Carlo interpretations of the theory. There is considerable scope to tailor the theory introduced in this paper to particular application settings (such as the big data setting), construct efficient parallelised schemes, understand the approximation and computational efficiencies of other such unification paradigms, and explore new theoretical and methodological directions.
- Published
- 2019
29. Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression
- Author
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Zhihua Su, Yi Yang, Guangyu Zhu, and Xin Chen
- Subjects
0301 basic medicine ,Statistics and Probability ,Applied Mathematics ,General Mathematics ,Asymptotic distribution ,Estimator ,Feature selection ,01 natural sciences ,Agricultural and Biological Sciences (miscellaneous) ,Oracle ,010104 statistics & probability ,03 medical and health sciences ,030104 developmental biology ,Bayesian multivariate linear regression ,Linear predictor function ,Linear regression ,Statistics ,Statistics::Methodology ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Invariant (mathematics) ,General Agricultural and Biological Sciences ,Mathematics - Abstract
The envelope model allows efficient estimation in multivariate linear regression. In this paper, we propose the sparse envelope model, which is motivated by applications where some response variables are invariant with respect to changes of the predictors and have zero regression coefficients. The envelope estimator is consistent but not sparse, and in many situations it is important to identify the response variables for which the regression coefficients are zero. The sparse envelope model performs variable selection on the responses and preserves the efficiency gains offered by the envelope model. Response variable selection arises naturally in many applications, but has not been studied as thoroughly as predictor variable selection. In this paper, we discuss response variable selection in both the standard multivariate linear regression and the envelope contexts. In response variable selection, even if a response has zero coefficients, it should still be retained to improve the estimation efficiency of the nonzero coefficients. This is different from the practice in predictor variable selection. We establish consistency and the oracle property and obtain the asymptotic distribution of the sparse envelope estimator.
- Published
- 2016
30. Group algebras and coding theory
- Author
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Marinês Guerreiro
- Subjects
Pure mathematics ,Group (mathematics) ,General Mathematics ,Minimum weight ,020206 networking & telecommunications ,Context (language use) ,0102 computer and information sciences ,02 engineering and technology ,Group algebra ,Coding theory ,01 natural sciences ,Algebra ,Finite field ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Statistics, Probability and Uncertainty ,Abelian group ,Mathematics - Abstract
Group algebras have been used in the context of Coding Theory since the beginning of the latter, but not in its full power. The article of Ferraz and Polcino Milies entitled Idempotents in group algebras and minimal abelian codes (Finite Fields Appl 13(2):382–393, 2007) gave origin to many thesis and papers linking these two subjects. In these works, the techniques of group algebras are mainly brought into play for the computing of the idempotents that generate the minimal codes and the minimum weight of such codes. In this paper I present a survey on the main results proceeding from applications of that seminal work.
- Published
- 2016
31. A note on the simulation of the Ginibre point process
- Author
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Laurent Decreusefond, Anaïs Vergne, Ian Flint, Data, Intelligence and Graphs (DIG), Laboratoire Traitement et Communication de l'Information (LTCI), Institut Mines-Télécom [Paris] (IMT)-Télécom Paris-Institut Mines-Télécom [Paris] (IMT)-Télécom Paris, Département Informatique et Réseaux (INFRES), Télécom ParisTech, Mathématiques discrètes, Codage et Cryptographie (MC2), and Réseaux, Mobilité et Services (RMS)
- Subjects
Statistics and Probability ,Property (philosophy) ,Distribution (number theory) ,General Mathematics ,02 engineering and technology ,point process simulation ,01 natural sciences ,Point process ,Computer Science::Hardware Architecture ,010104 statistics & probability ,Determinantal point process ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,60G60 ,Ginibre point process ,Plane (geometry) ,010102 general mathematics ,15A52 ,020206 networking & telecommunications ,Algebra ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,60K35 ,60G55 ,Statistics, Probability and Uncertainty ,Complex plane ,Random matrix - Abstract
The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.
- Published
- 2015
32. Relativization, absolutization, and latticization in Ring and Module Theory
- Author
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Toma Albu
- Subjects
Modular lattice ,Quotient category ,Grothendieck category ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Computational Theory and Mathematics ,Mathematics::Category Theory ,Semisimple module ,Torsion (algebra) ,Krull dimension ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this survey paper we illustrate a general strategy which consists on putting a module-theoretical result into a latticial frame (we call it latticization), in order to translate that result to Grothendieck categories (we call it absolutization) and module categories equipped with hereditary torsion theories (we call it relativization). The renowned Hopkins–Levitzki Theorem and Osofsky–Smith Theorem from Ring and Module Theory, we will discuss in the last two sections of the paper, are among the most relevant illustrations of the power of this strategy.
- Published
- 2015
33. Visualizing Tests for Equality of Covariance Matrices
- Author
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Michael Friendly and Matthew Sigal
- Subjects
FOS: Computer and information sciences ,Statistics and Probability ,Multivariate statistics ,General Mathematics ,05 social sciences ,Linear model ,050301 education ,Covariance ,01 natural sciences ,Plot (graphics) ,62-07, 62-09, 62H15 ,Methodology (stat.ME) ,010104 statistics & probability ,Multivariate analysis of variance ,Statistics ,Statistics::Methodology ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Graphics ,0503 education ,Statistics - Methodology ,Mathematics - Abstract
This paper explores a variety of topics related to the question of testing the equality of covariance matrices in multivariate linear models, particularly in the MANOVA setting. The main focus is on graphical methods that can be used to address the evaluation of this assumption. We introduce some extensions of data ellipsoids, hypothesis-error (HE) plots and canonical discriminant plots and demonstrate how they can be applied to the testing of equality of covariance matrices. Further, a simple plot of the components of Box's M test is proposed that shows _how_ groups differ in covariance and also suggests other visualizations and alternative test statistics. These methods are implemented and freely available in the **heplots** and **candisc** packages for R. Examples from the paper are available in supplementary materials., The American Statistician, in press (2018)
- Published
- 2018
34. Bridges with random length: Gamma case
- Author
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Mohamed Erraoui, Mohammed Louriki, and Astrid Hilbert
- Subjects
Statistics and Probability ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,Process (computing) ,Markov process ,Sense (electronics) ,01 natural sciences ,Natural filtration ,010104 statistics & probability ,symbols.namesake ,Bounded function ,FOS: Mathematics ,Filtration (mathematics) ,symbols ,Countable set ,0101 mathematics ,Statistics, Probability and Uncertainty ,Jump process ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.
- Published
- 2018
35. Maximum of Catalytic Branching Random Walk with Regularly Varying Tails
- Author
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Ekaterina Vl. Bulinskaya
- Subjects
Statistics and Probability ,education.field_of_study ,Laplace transform ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Population ,Probability (math.PR) ,Asymptotic distribution ,60J80, 60F05 ,Random walk ,01 natural sciences ,010104 statistics & probability ,Branching random walk ,FOS: Mathematics ,Large deviations theory ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Random variable ,Mathematics - Probability ,Mathematics - Abstract
For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to a non-trivial law. An explicit formula is provided for this normalization and non-linear integral equations are obtained to determine the limiting distribution function. The novelty consists in establishing the weak convergence for CBRW with "heavy" tails, in contrast to the known behavior in case of "light" tails of the random walk jumps. The new tools such as "many-to-few lemma" and spinal decomposition appear non-efficient here. The approach developed in the paper combines the techniques of renewal theory, Laplace transform, non-linear integral equations and large deviations theory for random sums of random variables. Keywords and phrases: catalytic branching random walk, heavy tails, regular varying tails, spread of population, L log L condition.
- Published
- 2018
- Full Text
- View/download PDF
36. One-sided FKPP travelling waves for homogeneous fragmentation processes
- Author
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Robert Knobloch and Publica
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Lévy process ,010104 statistics & probability ,One sided ,Homogeneous ,Traveling wave ,Jump ,Uniqueness ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Brownian motion ,Mathematics - Abstract
In this paper, we introduce an analogue of the classical one-sided FKPP equation in the context of homogeneous fragmentation processes. The main result of the present paper is concerned with the existence and uniqueness of one-sided FKPP travelling waves in this setting. In addition, we prove some analytic properties of such travelling waves. Our techniques make use of fragmentation processes with killing, an associated product martingale as well as various properties of Levy processes. This paper is mainly concerned with general fragmentation processes, but we also devote a section to related considerations regarding fragmentations with a finite dislocation measure, where we obtain a stronger result than for fragmentation processes with an infinite jump activity over finite time horizons. Furthermore, we discuss the relation of our problem to similar questions in the setting of branching Brownian motions, which provides a motivation for our approach.
- Published
- 2018
37. Signal-plus-noise matrix models: eigenvector deviations and fluctuations
- Author
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Joshua Cape, Minh Tang, and Carey E. Priebe
- Subjects
Statistics and Probability ,General Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Asymptotic distribution ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,010104 statistics & probability ,Matrix (mathematics) ,Stochastic block model ,62H12, 15A18, 15B52 ,0502 economics and business ,FOS: Mathematics ,Statistical physics ,0101 mathematics ,Eigenvalues and eigenvectors ,050205 econometrics ,Mathematics ,Random graph ,Applied Mathematics ,05 social sciences ,Probabilistic logic ,Agricultural and Biological Sciences (miscellaneous) ,Range (mathematics) ,Statistics, Probability and Uncertainty ,General Agricultural and Biological Sciences ,Random matrix - Abstract
Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science, and applied mathematics. This paper characterizes the behavior of perturbed eigenvectors for a range of signal-plus-noise matrix models encountered in both statistical and random matrix theoretic settings. We prove both first-order approximation results (i.e. sharp deviations) as well as second-order distributional limit theory (i.e. fluctuations). The concise methodology considered in this paper synthesizes tools rooted in two core concepts, namely (i) deterministic decompositions of matrix perturbations and (ii) probabilistic matrix concentration phenomena. We illustrate our theoretical results via simulation examples involving stochastic block model random graphs., Comment: 12 pages, 2 figures, 1 table
- Published
- 2018
- Full Text
- View/download PDF
38. Assumption Lean Regression
- Author
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Linda Zhao, Edward I. George, Lawrence D. Brown, Andreas Buja, Arun Kumar Kuchibhotla, Weijie J. Su, and Richard A. Berk
- Subjects
Statistics and Probability ,Generalized linear model ,FOS: Computer and information sciences ,General Mathematics ,05 social sciences ,01 natural sciences ,050105 experimental psychology ,Regression ,Methodology (stat.ME) ,010104 statistics & probability ,Linear regression ,Econometrics ,0501 psychology and cognitive sciences ,Observational study ,0101 mathematics ,Statistics, Probability and Uncertainty ,Statistics - Methodology ,Mathematics - Abstract
It is well known that models used in conventional regression analysis are commonly misspecified. A standard response is little more than a shrug. Data analysts invoke Box's maxim that all models are wrong and then proceed as if the results are useful nevertheless. In this paper, we provide an alternative. Regression models are treated explicitly as approximations of a true response surface that can have a number of desirable statistical properties, including estimates that are asymptotically unbiased. Valid statistical inference follows. We generalize the formulation to include regression functionals, which broadens substantially the range of potential applications. An empirical application is provided to illustrate the paper's key concepts., Comment: Submitted for review, 21 pages, 2 figures
- Published
- 2018
- Full Text
- View/download PDF
39. LARGE DEVIATIONS OF THE THRESHOLD ESTIMATOR OF INTEGRATED (CO-)VOLATILITY VECTOR IN THE PRESENCE OF JUMPS
- Author
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Hacène Djellout, Hui Jiang, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Clermont Auvergne [2017-2020] (UCA [2017-2020])-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [Nanjing], and Nanjing University of Aeronautics and Astronautics [Nanjing] (NUAA)
- Subjects
Statistics and Probability ,Threshold estimator ,Realized variance ,General Mathematics ,Quadratic variation ,Jump Poisson ,01 natural sciences ,Lévy process ,010104 statistics & probability ,Large deviation principle ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,0502 economics and business ,Econometrics ,Applied mathematics ,0101 mathematics ,050205 econometrics ,Mathematics ,[QFIN.GN]Quantitative Finance [q-fin]/General Finance [q-fin.GN] ,Stochastic process ,05 social sciences ,Estimator ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Jump ,Jump Poisson,Large deviation principle , Quadratic variation , Threshold estimator ,Large deviations theory ,Statistics, Probability and Uncertainty ,Financial econometrics ,Rate function ,Mathematics - Probability - Abstract
Recently a considerable interest has been paid on the estimation problem of the realized volatility and covolatility by using high-frequency data of financial price processes in financial econometrics. Threshold estimation is one of the useful techniques in the inference for jump-type stochastic processes from discrete observations. In this paper, we adopt the threshold estimator introduced by Mancini where only the variations under a given threshold function are taken into account. The purpose of this work is to investigate large and moderate deviations for the threshold estimator of the integrated variance-covariance vector. This paper is an extension of the previous work in Djellout et al. where the problem has been studied in absence of the jump component. We will use the approximation lemma to prove the LDP. As the reader can expect we obtain the same results as in the case without jump., Comment: 16pages
- Published
- 2018
40. Simple Conditions for Metastability of Continuous Markov Chains
- Author
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Oren Mangoubi, Aaron Smith, and Natesh S. Pillai
- Subjects
Statistics and Probability ,FOS: Computer and information sciences ,Markov chain ,General Mathematics ,010102 general mathematics ,Probability (math.PR) ,Multimodal distribution ,Random walk ,Statistics - Computation ,01 natural sciences ,Hybrid Monte Carlo ,010104 statistics & probability ,Mixing (mathematics) ,Metastability ,FOS: Mathematics ,Mixture distribution ,Spectral gap ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,Computation (stat.CO) ,Mathematics - Abstract
A family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit $\textit{metastable mixing}$ with $\textit{modes}$ $S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min(\Phi_{\beta}(S_{\beta}^{(1)}), \ldots, \Phi_{\beta}(S_{\beta}^{(k)}))$ of its modes. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis-Hastings chain targeting a mixture distribution. Our work differs from existing work on metastability in that, for the class of examples we are interested in, it gives an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp) while at the same time giving technical conditions that are easier to verify for many statistical examples. Our bounds from this paper are used in a companion paper to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions., Comment: arXiv admin note: text overlap with arXiv:1808.03230
- Published
- 2018
- Full Text
- View/download PDF
41. Partially informed investors: hedging in an incomplete market with default
- Author
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Paola Tardelli
- Subjects
Statistics and Probability ,exponential utility ,General Mathematics ,backward stochastic differential equation ,93E11 ,01 natural sciences ,default time ,Unobservable ,010104 statistics & probability ,Stochastic differential equation ,Order (exchange) ,Bellman equation ,Incomplete markets ,Econometrics ,49L20 ,Asset (economics) ,0101 mathematics ,Mathematics ,dynamic programming ,Stochastic control ,Actuarial science ,Optimal investment ,010102 general mathematics ,filtering ,93E03 ,Exponential utility ,Statistics, Probability and Uncertainty - Abstract
In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.
- Published
- 2015
42. The limiting failure rate for a convolution of life distributions
- Author
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Henry W. Block, Thomas H. Savits, and Naftali A. Langberg
- Subjects
failure rate function ,Statistics and Probability ,education.field_of_study ,decreasing failure rate ,Component (thermodynamics) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Population ,Block (permutation group theory) ,Monotonic function ,Failure rate ,Limiting ,Reliability ,01 natural sciences ,increasing failure rate ,Convolution ,62N05 ,010104 statistics & probability ,convolution ,60K10 ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.
- Published
- 2015
43. On the Semicircular Law of Large-Dimensional Random Quaternion Matrices
- Author
-
Zhidong Bai, Jiang Hu, and Yanqing Yin
- Subjects
Statistics and Probability ,Lemma (mathematics) ,Pure mathematics ,General Mathematics ,Gaussian ,010102 general mathematics ,01 natural sciences ,Hermitian matrix ,010104 statistics & probability ,symbols.namesake ,Law ,symbols ,Multiplication ,0101 mathematics ,Statistics, Probability and Uncertainty ,Quaternion ,Commutative property ,Eigenvalues and eigenvectors ,Symplectic geometry ,Mathematics - Abstract
It is well known that the Gaussian symplectic ensemble is defined on the space of \(n\times n\) quaternion self-dual Hermitian matrices with Gaussian random elements. There is a huge body of literature regarding this kind of matrices based on the exact known form of the density function of the eigenvalues (see Erdős in Russ Math Surv 66(3):507–626, 2011; Erdős in Probab Theory Relat Fields 154(1–2):341–407, 2012; Erdős et al. in Adv Math 229(3):1435–1515, 2012; Knowles and Yin in Probab Theory Relat Fields, 155(3–4):543–582, 2013; Tao and Vu in Acta Math 206(1):127–204, 2011; Tao and Vu in Electron J Probab 16(77):2104–2121, 2011). Due to the fact that multiplication of quaternions is not commutative, few works about large-dimensional quaternion self-dual Hermitian matrices are seen without normality assumptions. As in natural, we shall get more universal results by removing the Gaussian condition. For the first step, in this paper, we prove that the empirical spectral distribution of the common quaternion self-dual Hermitian matrices tends to the semicircular law. The main tool to establish the universal result is given as a lemma in this paper as well.
- Published
- 2015
44. Quasistochastic matrices and Markov renewal theory
- Author
-
Gerold Alsmeyer
- Subjects
Statistics and Probability ,Markov kernel ,General Mathematics ,perpetuity ,01 natural sciences ,age-dependent multitype branching process ,010104 statistics & probability ,Matrix (mathematics) ,random difference equation ,60K05 ,Markov renewal process ,Quasistochastic matrix ,60J45 ,Nonnegative matrix ,Renewal theory ,Markov renewal equation ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics ,Discrete mathematics ,Markov chain ,010102 general mathematics ,Stochastic matrix ,Stone-type decomposition ,60K15 ,Markov renewal theorem ,spread out ,60J10 ,Statistics, Probability and Uncertainty ,Markov random walk - Abstract
Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.
- Published
- 2014
45. Leaf Space Isometries of Singular Riemannian Foliations and Their Spectral Properties
- Author
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M. R. Sandoval and Ian M. Adelstein
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Quantitative Biology::Tissues and Organs ,General Mathematics ,Space (mathematics) ,01 natural sciences ,Mathematics - Spectral Theory ,Group action ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Spectral Theory (math.SP) ,Mathematics ,Mean curvature ,010102 general mathematics ,Spectral geometry ,Isospectral ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Isometry ,Vector field ,58J50, 58J53, 22D99, 53C12 ,Mathematics::Differential Geometry ,010307 mathematical physics ,Statistics, Probability and Uncertainty ,Laplace operator - Abstract
In this paper, the authors consider leaf spaces of singular Riemannian foliations $\mathcal{F}$ on compact manifolds $M$ and the associated $\mathcal{F}$-basic spectrum on $M$, $spec_B(M, \mathcal{F}),$ counted with multiplicities. Recently, a notion of smooth isometry $\varphi: M_1/\mathcal{F}_1\rightarrow M_2/\mathcal{F}_2$ between the leaf spaces of such singular Riemannian foliations $(M_1,\mathcal{F}_1)$ and $(M_2,\mathcal{F}_2)$ has appeared in the literature. In this paper, the authors provide an example to show that the existence a smooth isometry of leaf spaces as above is not sufficient to guarantee the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2).$ The authors then prove that if some additional conditions involving the geometry of the leaves are satisfied, then the equality of $spec_B(M_1,\mathcal{F}_1)$ and $spec_B(M_2,\mathcal{F}_2)$ is guaranteed. Consequences and applications to orbifold spectral theory, isometric group actions, and their reductions are also explored., Comment: 11 pages. arXiv admin note: text overlap with arXiv:1607.05593 This version corrects a missing hypothesis and includes a cleaner presentation with a shorter proof of the main result. An appendix has also been added with the original proof of one of the main results
- Published
- 2017
46. Infinite-server queues with Hawkes input
- Author
-
David Koops, Onno Boxma, Mayank Saxena, Michel Mandjes, Stochastic Operations Research, and Stochastics (KDV, FNWI)
- Subjects
Statistics and Probability ,Exponential distribution ,Distribution (number theory) ,General Mathematics ,0211 other engineering and technologies ,Markov process ,02 engineering and technology ,heavy traffic ,Branching process ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Joint probability distribution ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,heavy-tailed distribution ,Queue ,Self-exciting process ,Hawkes process ,Mathematics ,Heavy-tailed distribution heavy traffic ,021103 operations research ,Probability (math.PR) ,Process (computing) ,Heavy-tailed distribution ,symbols ,Statistics, Probability and Uncertainty ,infinite-server queue ,Mathematics - Probability - Abstract
In this paper we study the number of customers in infinite-server queues with a self-exciting (Hawkes) arrival process. Initially we assume that service requirements are exponentially distributed and that the Hawkes arrival process is of a Markovian nature. We obtain a system of differential equations that characterizes the joint distribution of the arrival intensity and the number of customers. Moreover, we provide a recursive procedure that explicitly identifies (transient and stationary) moments. Subsequently, we allow for non-Markovian Hawkes arrival processes and non-exponential service times. By viewing the Hawkes process as a branching process, we find that the probability generating function of the number of customers in the system can be expressed in terms of the solution of a fixed-point equation. We also include various asymptotic results: we derive the tail of the distribution of the number of customers for the case that the intensity jumps of the Hawkes process are heavy-tailed, and we consider a heavy-traffic regime. We conclude the paper by discussing how our results can be used computationally and by verifying the numerical results via simulations., v3: some minor corrections
- Published
- 2017
47. Asymptotic Properties of Approximate Bayesian Computation
- Author
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Gael M. Martin, Judith Rousseau, David T. Frazier, Christian P. Robert, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), University of Warwick [Coventry], Monash University [Melbourne], Université Paris Dauphine-PSL, and Université Paris sciences et lettres (PSL)
- Subjects
0301 basic medicine ,Statistics and Probability ,FOS: Computer and information sciences ,General Mathematics ,Asymptotic distribution ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,Statistics - Computation ,Posterior mean ,Methodology (stat.ME) ,010104 statistics & probability ,03 medical and health sciences ,Simple (abstract algebra) ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,FOS: Mathematics ,Applied mathematics ,Statistics::Methodology ,Statistical analysis ,0101 mathematics ,QA ,ComputingMilieux_MISCELLANEOUS ,Computation (stat.CO) ,Statistics - Methodology ,Bernstein–von Mises theorem ,Mathematics ,Applied Mathematics ,[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH] ,Agricultural and Biological Sciences (miscellaneous) ,[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation ,Statistics::Computation ,Identification (information) ,030104 developmental biology ,Econometric and statistical methods ,Statistics, Probability and Uncertainty ,Approximate Bayesian computation ,Econometrics not elsewhere classified ,General Agricultural and Biological Sciences ,Focus (optics) ,[STAT.ME]Statistics [stat]/Methodology [stat.ME] - Abstract
Approximate Bayesian computation allows for statistical analysis in models with intractable likelihoods. In this paper we consider the asymptotic behaviour of the posterior distribution obtained by this method. We give general results on the rate at which the posterior distribution concentrates on sets containing the true parameter, its limiting shape, and the asymptotic distribution of the posterior mean. These results hold under given rates for the tolerance used within the method, mild regularity conditions on the summary statistics, and a condition linked to identification of the true parameters. Implications for practitioners are discussed., This 31 pages paper is a revised version of the paper, including supplementary material
- Published
- 2017
48. How Principled and Practical Are Penalised Complexity Priors?
- Author
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Judith Rousseau, Christian P. Robert, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and University of Warwick [Coventry]
- Subjects
Statistics and Probability ,business.industry ,General Mathematics ,Bayesian probability ,Overfitting ,16. Peace & justice ,Base (topology) ,[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation ,01 natural sciences ,010104 statistics & probability ,03 medical and health sciences ,0302 clinical medicine ,Component (UML) ,Prior probability ,Artificial intelligence ,0101 mathematics ,Statistics, Probability and Uncertainty ,business ,030217 neurology & neurosurgery ,Mathematics - Abstract
International audience; This note discusses the paper "Penalising model component complexity" by Simpson et al. (2017). While we acknowledge the highly novel approach to prior construction and commend the authors for setting new-encompassing principles that will Bayesian modelling, and while we perceive the potential connection with other branches of the literature, we remain uncertain as to what extent the principles exposed in the paper can be developed outside specific models, given their lack of precision. The very notions of model component, base model, overfitting prior are for instance conceptual rather than mathematical and we thus fear the concept of penalised complexity may not further than extending first-guess priors into larger families, thus failing to establish reference priors on a novel sound ground.
- Published
- 2017
49. Improved availability bounds for binary and multistate monotone systems with independent component processes
- Author
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Bent Natvig and Jørund Gåsemyr
- Subjects
Statistics and Probability ,Reliability theory ,Mathematical optimization ,General Mathematics ,010102 general mathematics ,Binary number ,Interval (mathematics) ,01 natural sciences ,Upper and lower bounds ,010104 statistics & probability ,Monotone polygon ,Component (UML) ,Path (graph theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Time point ,Mathematics - Abstract
Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities of the system in an interval, i.e. the probabilities that the system performs above the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities exactly, but if the component performance processes are independent, it is possible to construct lower bounds based on the component availabilities to the different levels over the interval. In this paper we show that by treating the component availabilities over the interval as if they were availabilities at a single time point, we obtain an improved lower bound. Unlike previously given bounds, the new bound does not require the identification of all minimal path or cut vectors.
- Published
- 2017
50. Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables
- Author
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Ruodu Wang
- Subjects
Statistics and Probability ,General Mathematics ,Structure (category theory) ,Value (computer science) ,91E30 ,01 natural sciences ,value at risk ,Combinatorics ,010104 statistics & probability ,0502 economics and business ,60E05 ,Limit (mathematics) ,0101 mathematics ,Mathematics ,Discrete mathematics ,050208 finance ,05 social sciences ,Expected shortfall ,Distribution (mathematics) ,Dependence bound ,complete mixability ,modeling uncertainty ,60E15 ,Marginal distribution ,Statistics, Probability and Uncertainty ,Random variable ,Value at risk - Abstract
Suppose that X 1, …, X n are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X 1 + · · · + X n < s) over all possible dependence structures, denoted by m n,F (s). We show that m n,F (ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of m n,F (ns) for any s ∈ R with an error of at most n -1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.
- Published
- 2014
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