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Search Results
52. Flooding and diameter in general weighted random graphs
- Author
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Thomas Mountford and Jacques Saliba
- Subjects
Statistics and Probability ,General Mathematics ,media_common.quotation_subject ,flooding time ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,0101 mathematics ,diameter ,Mathematics ,media_common ,60C05, 05C80, 90B15 ,Random graph ,first passage percolation ,Probability (math.PR) ,010102 general mathematics ,1st passage percolation ,First passage percolation ,Infinity ,continuous branching process ,configuration model ,Flooding (computer networking) ,Exponential function ,Vertex (geometry) ,Weight distribution ,Graph (abstract data type) ,Statistics, Probability and Uncertainty ,Mathematics - Probability ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper we study first passage percolation on a random graph model, the configuration model. We first introduce the notions of weighted diameter, which is the maximum of the weighted lengths of all optimal paths between any two vertices in the graph, and the flooding time, which represents the time (weighted length) needed to reach all the vertices in the graph starting from a uniformly chosen vertex. Our result consists in describing the asymptotic behavior of the diameter and the flooding time, as the number of vertices n tends to infinity, in the case where the weight distribution G has an exponential tail behavior, and proving that this category of distributions is the largest possible for which the asymptotic behavior holds.
- Published
- 2020
53. On the behavior of the failure rate and reversed failure rate in engineering systems
- Author
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Mahdi Tavangar
- Subjects
Statistics and Probability ,021103 operations research ,General Mathematics ,Cumulative distribution function ,Order statistic ,0211 other engineering and technologies ,Failure rate ,02 engineering and technology ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,System failure ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Reliability (statistics) ,Mathematics - Abstract
In this paper the behaviour of the failure rate and reversed failure rate of an n-component coherent system is studied, where it is assumed that the lifetimes of the components are independent and have a common cumulative distribution function F. Sufficient conditions are provided under which the system failure rate is increasing and the corresponding reversed failure rate is decreasing. We also study the stochastic and ageing properties of doubly truncated random variables for coherent systems.
- Published
- 2020
54. On the extension of signature-based representations for coherent systems with dependent non-exchangeable components
- Author
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Jorge Navarro and Juan Fernández-Sánchez
- Subjects
Statistics and Probability ,Discrete mathematics ,Independent and identically distributed random variables ,Class (set theory) ,General Mathematics ,Reliability (computer networking) ,010102 general mathematics ,Copula (linguistics) ,Extension (predicate logic) ,01 natural sciences ,Signature (logic) ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Representation (mathematics) ,Mathematics - Abstract
The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.
- Published
- 2020
55. Optimal stopping for measure-valued piecewise deterministic Markov processes
- Author
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Maud Joubaud, Bertrand Cloez, Benoîte de Saporta, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie (MISTEA), Institut National de la Recherche Agronomique (INRA)-Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), Region Occitanie, Region Ile-de-France, French National Research Agency (ANR), Institut national d’études supérieures agronomiques de Montpellier (Montpellier SupAgro)-Institut National de la Recherche Agronomique (INRA), and de Saporta, Benoîte
- Subjects
Statistics and Probability ,[MATH.MATH-PR] Mathematics [math]/Probability [math.PR] ,Mathematical optimization ,General Mathematics ,Population ,Markov process ,01 natural sciences ,Measure (mathematics) ,measure space ,010104 statistics & probability ,03 medical and health sciences ,symbols.namesake ,Bellman equation ,FOS: Mathematics ,population dynamics ,Optimal stopping ,0101 mathematics ,education ,030304 developmental biology ,Mathematics ,dynamic programming ,0303 health sciences ,Sequence ,education.field_of_study ,Markov processes ,Probability (math.PR) ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Dynamic programming ,optimal stopping ,symbols ,Piecewise ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.
- Published
- 2020
56. Generalized stacked contact process with variable host fitness
- Author
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Eric Foxall and Nicolas Lanchier
- Subjects
Statistics and Probability ,Contact process ,education.field_of_study ,Host (biology) ,General Mathematics ,Probability (math.PR) ,Population ,Integer lattice ,Forest-fire model ,Spin system ,01 natural sciences ,010305 fluids & plasmas ,Variable (computer science) ,60K35 ,Spatial model ,0103 physical sciences ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,Biological system ,education ,Mathematics - Probability ,Mathematics - Abstract
The stacked contact process is a three-state spin system that describes the co-evolution of a population of hosts together with their symbionts. In a nutshell, the hosts evolve according to a contact process while the symbionts evolve according to a contact process on the dynamic subset of the lattice occupied by the host population, indicating that the symbiont can only live within a host. This paper is concerned with a generalization of this system in which the symbionts may affect the fitness of the hosts by either decreasing (pathogen) or increasing (mutualist) their birth rate. Standard coupling arguments are first used to compare the process with other interacting particle systems and deduce the long-term behavior of the host-symbiont system in several parameter regions. The mean-field approximation of the process is also studied in detail and compared to the spatial model. Our main result focuses on the case where unassociated hosts have a supercritical birth rate whereas hosts associated to a pathogen have a subcritical birth rate. In this case, the mean-field model predicts coexistence of the hosts and their pathogens provided the infection rate is large enough. For the spatial model, however, only the hosts survive on the one-dimensional integer lattice., 23 pages, 4 figures
- Published
- 2020
57. The alpha-mixture of survival functions
- Author
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Nader Ebrahimi, Ehsan S. Soofi, and Majid Asadi
- Subjects
Statistics and Probability ,Property (philosophy) ,General Mathematics ,020206 networking & telecommunications ,Failure rate ,02 engineering and technology ,01 natural sciences ,Stochastic ordering ,Interpretation (model theory) ,010104 statistics & probability ,Alpha (programming language) ,Monotone polygon ,Constant elasticity of substitution ,0202 electrical engineering, electronic engineering, information engineering ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper presents a flexible family which we call the $\alpha$ -mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The $\alpha$ -mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for $\alpha$ . We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.
- Published
- 2019
58. Persistence probability of a random polynomial arising from evolutionary game theory
- Author
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Viet Viet Hung Pham, Van Hao Can, and Manh Hong Duong
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Statistics and Probability ,Computer Science::Computer Science and Game Theory ,General Mathematics ,Evolutionary game theory ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Applied mathematics ,Asymptotic formula ,Mathematics - Dynamical Systems ,0101 mathematics ,Quantitative Biology - Populations and Evolution ,Real line ,Gaussian process ,Mathematical Physics ,Mathematics ,Equilibrium point ,Sequence ,Probability (math.PR) ,010102 general mathematics ,Populations and Evolution (q-bio.PE) ,Mathematical Physics (math-ph) ,FOS: Biological sciences ,symbols ,Key (cryptography) ,Statistics, Probability and Uncertainty ,Persistence (discontinuity) ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
In this paper, we obtain an asymptotic formula for the persistence probability in the positive real line of a random polynomial arising from evolutionary game theory. It corresponds to the probability that a multi-player two-strategy random evolutionary game has no internal equilibria. The key ingredient is to approximate the sequence of random polynomials indexed by their degrees by an appropriate centered stationary Gaussian process., revised version
- Published
- 2019
59. Sums of standard uniform random variables
- Author
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Bin Wang, Tiantian Mao, and Ruodu Wang
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Statistics and Probability ,050208 finance ,Uniform distribution (continuous) ,General Mathematics ,Probability (math.PR) ,05 social sciences ,Aggregate (data warehouse) ,Characterization (mathematics) ,Type (model theory) ,01 natural sciences ,Set (abstract data type) ,010104 statistics & probability ,Dimension (vector space) ,0502 economics and business ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Marginal distribution ,Random variable ,Mathematics - Probability ,Mathematics - Abstract
In this paper, we analyze the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. An analytical result with full generality is not available in this case, and it seems to be out of reach with existing techniques. For dimension greater or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.
- Published
- 2019
60. The De Vylder–Goovaerts conjecture holds within the diffusion limit
- Author
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Nabil Kazi-Tani, Stefan Ankirchner, Christophette Blanchet-Scalliet, Institut für Mathematik, Friedrich-Schiller-Universität = Friedrich Schiller University Jena [Jena, Germany], Probabilités, statistique, physique mathématique (PSPM), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Sciences Actuarielle et Financière (SAF), Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon
- Subjects
Statistics and Probability ,Approximations of π ,Ruin probability ,General Mathematics ,Open problem ,[QFIN.RM]Quantitative Finance [q-fin]/Risk Management [q-fin.RM] ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,0502 economics and business ,Risk theory ,Applied mathematics ,0101 mathematics ,Diffusion (business) ,Gaussian process ,Mathematics ,050208 finance ,Conjecture ,05 social sciences ,Heavy traffic approximation ,Diffusion approximations ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Equalized claims ,Distribution (mathematics) ,Jump ,symbols ,Statistics, Probability and Uncertainty - Abstract
The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite-time ruin probability in a standard risk model is greater than or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T.In this paper, we consider the diffusion approximations of both the standard risk model and its associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a Gaussian process satisfying a simple SDE. We then compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. We conclude that the De Vylder and Goovaerts conjecture holds for the diffusion limits.
- Published
- 2019
61. The N-star network evolution model
- Author
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István Fazekas, Attila Perecsényi, and Csaba Noszály
- Subjects
Statistics and Probability ,Star network ,Random graph ,Discrete mathematics ,General Mathematics ,010102 general mathematics ,Joins ,Preferential attachment ,01 natural sciences ,Power law ,Doob–Meyer decomposition theorem ,010104 statistics & probability ,Asymptotic power ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Mathematics - Abstract
A new network evolution model is introduced in this paper. The model is based on cooperations of N units. The units are the nodes of the network and the cooperations are indicated by directed links. At each evolution step N units cooperate, which formally means that they form a directed N-star subgraph. At each step either a new unit joins the network and it cooperates with N − 1 old units, or N old units cooperate. During the evolution both preferential attachment and uniform choice are applied. Asymptotic power law distributions are obtained both for in-degrees and for out-degrees.
- Published
- 2019
62. Preservation of the mean residual life order for coherent and mixed systems
- Author
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Bo Henry Lindqvist, Nana Wang, and Francisco J. Samaniego
- Subjects
Statistics and Probability ,021103 operations research ,Component (thermodynamics) ,General Mathematics ,Order statistic ,0211 other engineering and technologies ,Closure (topology) ,02 engineering and technology ,Residual ,01 natural sciences ,Stochastic ordering ,Signature (logic) ,010104 statistics & probability ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Mathematics ,Counterexample - Abstract
The signature of a coherent system has been studied extensively in the recent literature. Signatures are particularly useful in the comparison of coherent or mixed systems under a variety of stochastic orderings. Also, certain signature-based closure and preservation theorems have been established. For example, it is now well known that certain stochastic orderings are preserved from signatures to system lifetimes when components have independent and identical distributions. This applies to the likelihood ratio order, the hazard rate order, and the stochastic order. The point of departure of the present paper is the question of whether or not a similar preservation result will hold for the mean residual life order. A counterexample is provided which shows that the answer is negative. Classes of distributions for the component lifetimes for which the latter implication holds are then derived. Connections to the theory of order statistics are also considered.
- Published
- 2019
63. A unified stability theory for classical and monotone Markov chains
- Author
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Takashi Kamihigashi and John Stachurski
- Subjects
Statistics and Probability ,Markov chain ,General Mathematics ,05 social sciences ,Stochastic dominance ,01 natural sciences ,Stability (probability) ,010104 statistics & probability ,Total variation ,Monotone polygon ,Stability theory ,0502 economics and business ,Metric (mathematics) ,Applied mathematics ,050207 economics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Probability measure - Abstract
In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.
- Published
- 2019
64. Variance estimates for random disc-polygons in smooth convex discs
- Author
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Ferenc Fodor and Viktor Vígh
- Subjects
Statistics and Probability ,Mathematics::Complex Variables ,General Mathematics ,010102 general mathematics ,Regular polygon ,Boundary (topology) ,Metric Geometry (math.MG) ,Variance (accounting) ,Computer Science::Computational Geometry ,01 natural sciences ,Probability model ,Combinatorics ,010104 statistics & probability ,Mathematics - Metric Geometry ,Intersection ,FOS: Mathematics ,Mathematics::Metric Geometry ,Astrophysics::Earth and Planetary Astrophysics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Astrophysics::Galaxy Astrophysics ,Inscribed figure ,52A22, 60D05 ,Mathematics - Abstract
In this paper we prove asymptotic upper bounds on the variance of the number of vertices and the missed area of inscribed random disc-polygons in smooth convex discs whose boundary isC+2. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles.
- Published
- 2018
65. Convergence rates for estimators of geodesic distances and Frèchet expectations
- Author
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Olivier Bodart, Catherine Aaron, Aaron, Catherine, Laboratoire de Mathématiques Blaise Pascal (LMBP), Université Blaise Pascal - Clermont-Ferrand 2 (UBP)-Centre National de la Recherche Scientifique (CNRS), Université Clermont Auvergne [2017-2020] (UCA [2017-2020]), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Statistics and Probability ,Geodesic distance ,Pure mathematics ,Geodesic ,Geometric inference ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,Estimator ,Boundary (topology) ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,010104 statistics & probability ,Rate of convergence ,62-07, 62G05, 62G20, 62H99 ,[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST] ,Convergence (routing) ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Fréchet expectations ,Statistics on manifolds ,[MATH.MATH-ST] Mathematics [math]/Statistics [math.ST] ,Mathematics - Abstract
Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.
- Published
- 2018
66. On optimal periodic dividend and capital injection strategies for spectrally negative Lévy models
- Author
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Kazutoshi Yamazaki, José-Luis Pérez, Kouji Yano, and Kei Noba
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,05 social sciences ,Poisson distribution ,01 natural sciences ,Arrival time ,Dividend payment ,Scale function ,010104 statistics & probability ,symbols.namesake ,Capital injection ,0502 economics and business ,Econometrics ,Reflection (physics) ,symbols ,Dividend ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
De Finetti’s optimal dividend problem has recently been extended to the case when dividend payments can be made only at Poisson arrival times. In this paper we consider the version with bail-outs where the surplus must be nonnegative uniformly in time. For a general spectrally negative Lévy model, we show the optimality of a Parisian-classical reflection strategy that pays the excess above a given barrier at each Poisson arrival time and also reflects from below at 0 in the classical sense.
- Published
- 2018
67. Uniform decomposition of probability measures: quantization, clustering and rate of convergence
- Author
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Julien Chevallier, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Statistique pour le Vivant et l’Homme (SVH), Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and ANR-11-LABX-0023,MME-DII,Modèles Mathématiques et Economiques de la Dynamique, de l'Incertitude et des Interactions(2011)
- Subjects
Statistics and Probability ,Approximations of π ,Computer Science::Information Retrieval ,General Mathematics ,Quantization (signal processing) ,010103 numerical & computational mathematics ,Rate of convergence ,Mathematical Subject Classification: 60E15, 62E17, 60B10, 60F99 ,01 natural sciences ,Minimax approximation algorithm ,Clustering ,Uniform approximation ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,010104 statistics & probability ,Quantization ,Applied mathematics ,Wasserstein distance ,0101 mathematics ,Statistics, Probability and Uncertainty ,Cluster analysis ,Probability measure ,Mathematics - Abstract
The study of finite approximations of probability measures has a long history. In Xu and Berger (2017), the authors focused on constrained finite approximations and, in particular, uniform ones in dimensiond=1. In the present paper we give an elementary construction of a uniform decomposition of probability measures in dimensiond≥1. We then use this decomposition to obtain upper bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained by Xu and Berger (2017) and to be sharp for generic probability measures.
- Published
- 2018
68. The degree-wise effect of a second step for a random walk on a graph
- Author
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Kenneth S. Berenhaut, Elizabeth J. Krizay, Hongyi Jiang, and Katelyn M. McNab
- Subjects
Statistics and Probability ,Combinatorics ,010104 statistics & probability ,Degree (graph theory) ,General Mathematics ,010102 general mathematics ,Friendship paradox ,Graph (abstract data type) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random walk ,01 natural sciences ,Mathematics - Abstract
In this paper we consider the degree-wise effect of a second step for a random walk on a graph. We prove that under the configuration model, for any fixed degree sequence the probability of exceeding a given degree threshold is smaller after two steps than after one. This builds on recent work of Krameret al.(2016) regarding the friendship paradox under random walks.
- Published
- 2018
69. A conditional limit theorem for high-dimensional ℓᵖ-spheres
- Author
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Kavita Ramanan and Steven Soojin Kim
- Subjects
Statistics and Probability ,Pure mathematics ,Kullback–Leibler divergence ,Convex geometry ,Euclidean space ,General Mathematics ,Convex set ,01 natural sciences ,010101 applied mathematics ,010104 statistics & probability ,Probability theory ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Rate function ,Event (probability theory) ,Mathematics - Abstract
The study of high-dimensional distributions is of interest in probability theory, statistics, and asymptotic convex geometry, where the object of interest is the uniform distribution on a convex set in high dimensions. The ℓp-spaces and norms are of particular interest in this setting. In this paper we establish a limit theorem for distributions on ℓp-spheres, conditioned on a rare event, in a high-dimensional geometric setting. As part of our proof, we establish a certain large deviation principle that is also relevant to the study of the tail behavior of random projections of ℓp-balls in a high-dimensional Euclidean space.
- Published
- 2018
70. Multi-point correlations for two-dimensional coalescing or annihilating random walks
- Author
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Oleg Zaboronski, Roger Tribe, and James Lukins
- Subjects
Statistics and Probability ,Particle system ,Logarithm ,Differential equation ,General Mathematics ,010102 general mathematics ,Field (mathematics) ,Simple random sample ,Random walk ,01 natural sciences ,010104 statistics & probability ,Correlation function ,Initial value problem ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we consider an infinite system of instantaneously coalescing rate 1 simple symmetric random walks on ℤ2, started from the initial condition with all sites in ℤ2 occupied. Two-dimensional coalescing random walks are a `critical' model of interacting particle systems: unlike coalescence models in dimension three or higher, the fluctuation effects are important for the description of large-time statistics in two dimensions, manifesting themselves through the logarithmic corrections to the `mean field' answers. Yet the fluctuation effects are not as strong as for the one-dimensional coalescence, in which case the fluctuation effects modify the large time statistics at the leading order. Unfortunately, unlike its one-dimensional counterpart, the two-dimensional model is not exactly solvable, which explains a relative scarcity of rigorous analytic answers for the statistics of fluctuations at large times. Our contribution is to find, for any N≥2, the leading asymptotics for the correlation functions ρN(x1,…,xN) as t→∞. This generalises the results for N=1 due to Bramson and Griffeath (1980) and confirms a prediction in the physics literature for N>1. An analogous statement holds for instantaneously annihilating random walks. The key tools are the known asymptotic ρ1(t)∼logt∕πt due to Bramson and Griffeath (1980), and the noncollision probability 𝒑NC(t), that no pair of a finite collection of N two-dimensional simple random walks meets by time t, whose asymptotic 𝒑NC(t)∼c0(logt)-(N2) was found by Cox et al. (2010). We re-derive the asymptotics, and establish new error bounds, both for ρ1(t) and 𝒑NC(t) by proving that these quantities satisfy effective rate equations; that is, approximate differential equations at large times. This approach can be regarded as a generalisation of the Smoluchowski theory of renormalised rate equations to multi-point statistics.
- Published
- 2018
71. New nonparametric classes of distributions in terms of mean time to failure in age replacement
- Author
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Baha-Eldin Khaledi, Muhyiddin Izadi, and Maryam Sharafi
- Subjects
Statistics and Probability ,Mean time between failures ,021103 operations research ,General Mathematics ,0211 other engineering and technologies ,Nonparametric statistics ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,010104 statistics & probability ,Statistics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The mean time to failure (MTTF) function in age replacement is used to evaluate the performance and effectiveness of the age replacement policy. In this paper, based on the MTTF function, we introduce two new nonparametric classes of lifetime distributions with nonmonotonic mean time to failure in age replacement; increasing then decreasing MTTF (IDMTTF) and decreasing then increasing MTTF (DIMTTF). The implications between these classes of distributions and some existing classes of nonmonotonic ageing classes are studied. The characterizations of IDMTTF and DIMTTF in terms of the scaled total time on test transform are also obtained.
- Published
- 2018
72. Reliability modeling of coherent systems with shared components based on sequential order statistics
- Author
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Somayeh Ashrafi, Somayeh Zarezadeh, and Majid Asadi
- Subjects
Statistics and Probability ,021103 operations research ,Dependency (UML) ,Signature matrix ,General Mathematics ,Order statistic ,0211 other engineering and technologies ,02 engineering and technology ,Function (mathematics) ,01 natural sciences ,Stochastic ordering ,Set (abstract data type) ,010104 statistics & probability ,Mixing (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,Reliability (statistics) ,Mathematics - Abstract
In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.
- Published
- 2018
73. Reach of repulsion for determinantal point processes in high dimensions
- Author
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François Baccelli and Eliza O'Reilly
- Subjects
Statistics and Probability ,Boolean model ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Radius ,Palm calculus ,01 natural sciences ,Measure (mathematics) ,Point process ,010104 statistics & probability ,Distribution (mathematics) ,Point (geometry) ,Determinantal point process ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ0,! such that there exists a point process η where Φ=Φ0,!∪η in distribution and Φ0,!∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n→∞. We also prove that for some DPPs, there exists an R∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √nR∗. This R∗ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high-dimensional Boolean models is given.
- Published
- 2018
74. Stochastic comparisons of coherent systems under different random environments
- Author
-
Yiying Zhang, Ebrahim Amini-Seresht, and Narayanaswamy Balakrishnan
- Subjects
Statistics and Probability ,Distortion function ,Mathematical optimization ,021103 operations research ,General Mathematics ,Hazard ratio ,0211 other engineering and technologies ,02 engineering and technology ,Sense (electronics) ,01 natural sciences ,Stochastic ordering ,010104 statistics & probability ,Work (electrical) ,Random environment ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.
- Published
- 2018
75. Asymptotic behavior for the Robbins–Monro process
- Author
-
Yu Miao and Manru Dong
- Subjects
Statistics and Probability ,010104 statistics & probability ,General Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Process (computing) ,Applied mathematics ,020201 artificial intelligence & image processing ,02 engineering and technology ,Moderate deviations ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
In this paper we study the Robbins–Monro procedure Xn+1 = Xn - an-1Yn with some fixed number a > 0 and establish the moderate deviation principle of the process {Xn}.
- Published
- 2018
76. Recovering a hidden community beyond the Kesten–Stigum threshold in O(|E|log*|V|) time
- Author
-
Yihong Wu, Jiaming Xu, and Bruce Hajek
- Subjects
Statistics and Probability ,Sublinear function ,General Mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Function (mathematics) ,Binary logarithm ,Belief propagation ,01 natural sciences ,Iterated logarithm ,Combinatorics ,010104 statistics & probability ,Stochastic block model ,Power iteration ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Statistics, Probability and Uncertainty ,Time complexity ,Mathematics - Abstract
Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime n1-o(1) ≤ K ≤ o(n). A critical parameter is the effective signal-to-noise ratio λ = K2(p - q)2 / ((n - K)q), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log*n + O(1) iterations, with the total time complexity O(|E|log*n), where log*n is the iterated logarithm of n. Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying a power iteration to the nonbacktracking matrix of the graph is shown to attain weak recovery if and only if λ > 1. In addition, the BP algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all K ≥ (n / logn) (ρBP + o(1)), where ρBP is a function of p / q.
- Published
- 2018
77. Hazard rate ordering of the largest order statistics from geometric random variables
- Author
-
Jeongsim Kim and Bara Kim
- Subjects
Statistics and Probability ,021103 operations research ,General Mathematics ,Open problem ,Order statistic ,Hazard ratio ,0211 other engineering and technologies ,Value (computer science) ,02 engineering and technology ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,Order (group theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
Mao and Hu (2010) left an open problem about the hazard rate order between the largest order statistics from two samples of n geometric random variables. Du et al. (2012) solved this open problem when n = 2, and Wang (2015) solved for 2 ≤ n ≤ 9. In this paper we completely solve this problem for any value of n.
- Published
- 2018
78. A quenched central limit theorem for biased random walks on supercritical Galton–Watson trees
- Author
-
Adam Bowditch
- Subjects
Statistics and Probability ,Galton watson ,Conjecture ,Invariance principle ,General Mathematics ,010102 general mathematics ,Random walk ,01 natural sciences ,Upper and lower bounds ,Supercritical fluid ,Combinatorics ,010104 statistics & probability ,Tree (set theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Central limit theorem - Abstract
In this paper we prove a quenched functional central limit theorem for a biased random walk on a supercritical Galton–Watson tree with leaves. This extends a result of Peres and Zeitouni (2008) where the case without leaves was considered. A conjecture of Ben Arous and Fribergh (2016) suggests an upper bound on the bias which we observe to be sharp.
- Published
- 2018
79. A temporal approach to the Parisian risk model
- Author
-
Jeff T.Y. Wong, Bin Li, and Gordon E. Willmot
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,05 social sciences ,Variation (game tree) ,01 natural sciences ,Lévy process ,010104 statistics & probability ,Risk model ,Bounded function ,Scheme (mathematics) ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we propose a new approach to study the Parisian ruin problem for spectrally negative Lévy processes. Since our approach is based on a hybrid observation scheme switching between discrete and continuous observations, we call it a temporal approach as opposed to the spatial approximation approach in the literature. Our approach leads to a unified proof for the underlying processes with bounded or unbounded variation paths, and our result generalizes Loeffen et al. (2013).
- Published
- 2018
80. Structure-preserving equivalent martingale measures for ℋ-SII models
- Author
-
David Criens
- Subjects
Statistics and Probability ,Pure mathematics ,050208 finance ,Integrable system ,General Mathematics ,05 social sciences ,01 natural sciences ,010104 statistics & probability ,Semimartingale ,Conditional independence ,0502 economics and business ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Mathematics - Abstract
In this paper we relate the set of structure-preserving equivalent martingale measures ℳsp for financial models driven by semimartingales with conditionally independent increments to a set of measurable and integrable functions 𝒴. More precisely, we prove that ℳsp ≠ ∅ if and only if 𝒴 ≠ ∅, and connect the sets ℳsp and 𝒴 to the semimartingale characteristics of the driving process. As examples we consider integrated Lévy models with independent stochastic factors and time-changed Lévy models and derive mild conditions for ℳsp ≠ ∅.
- Published
- 2018
81. On stochastic comparisons of k-out-of-n systems with Weibull components
- Author
-
Narayanaswamy Balakrishnan, Abedin Haidari, and Ghobad Barmalzan
- Subjects
Statistics and Probability ,021103 operations research ,Scale (ratio) ,General Mathematics ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Applied mathematics ,Order (group theory) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Weibull distribution - Abstract
In this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons of k-out-of-n systems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhao et al. (2016).
- Published
- 2018
82. Equivalent representations of max-stable processes via ℓp-norms
- Author
-
Marco Oesting
- Subjects
Statistics and Probability ,Pointwise ,Infinite number ,Pure mathematics ,Spectral representation ,010504 meteorology & atmospheric sciences ,Stochastic process ,General Mathematics ,Ergodicity ,Tail dependence ,01 natural sciences ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Extreme value theory ,Maxima ,0105 earth and related environmental sciences ,Mathematics - Abstract
While max-stable processes are typically written as pointwise maxima over an infinite number of stochastic processes, in this paper, we consider a family of representations based on ℓp-norms. This family includes both the construction of the Reich–Shaby model and the classical spectral representation by de Haan (1984) as special cases. As the representation of a max-stable process is not unique, we present formulae to switch between different equivalent representations. We further provide a necessary and sufficient condition for the existence of an ℓp-norm-based representation in terms of the stable tail dependence function of a max-stable process. Finally, we discuss several properties of the represented processes such as ergodicity or mixing.
- Published
- 2018
83. Multiple drawing multi-colour urns by stochastic approximation
- Author
-
Olfa Selmi, Cécile Mailler, and Nabil Lasmar
- Subjects
Statistics and Probability ,Mathematics(all) ,General Mathematics ,Markov process ,Stochastic approximation ,01 natural sciences ,Multiple drawing Pólya urn ,010104 statistics & probability ,symbols.namesake ,Polya urn ,stochastic approximation ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Discrete mathematics ,discrete-time martingale ,Probability (math.PR) ,010102 general mathematics ,Ball (bearing) ,symbols ,limit theorem ,Statistics, Probability and Uncertainty ,reinforced process ,Mathematics - Probability - Abstract
A classical P��lya urn scheme is a Markov process whose evolution is encoded by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete time-step, we draw a ball uniformly at random, denote its colour $c$, and replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all $1\leq j\leq d$). We are interested in multi-drawing P��lya urns, where the replacement rule depends on the random drawing of a set of $m$ balls from the urn (with or without replacement). This generalisation has already been studied in the literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053), where second order asymptotic results are proved for $2$-colour urns under the balanced and the affinity assumptions. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to remove the affinity hypothesis of Kuba & Mahmoud and generalise the result to more-than-two-colour urns. We also give some partial results in the two-colour non-balanced case., This new arxiv version (v6) corrects a mistake that we discovered in the previous versions of this paper (v1-5). The mistake was in Theorem 1$(a)$ and in the last sentence of Theorem 4. In this new version, Theorem 1$(a)$ has been corrected, and Theorem 4 has been deleted
- Published
- 2018
84. Uniformly efficient simulation for extremes of Gaussian random fields
- Author
-
Gongjun Xu and Xiaoou Li
- Subjects
Statistics and Probability ,Random field ,General Mathematics ,Gaussian ,Probability (math.PR) ,010102 general mathematics ,Hölder condition ,01 natural sciences ,General family ,Gaussian random field ,010104 statistics & probability ,symbols.namesake ,Efficient estimator ,FOS: Mathematics ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,Mathematics - Probability ,Importance sampling ,Event (probability theory) ,Mathematics - Abstract
In this paper we consider the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problems. To overcome this issue, we propose a uniformly efficient estimator for a general family of Hölder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.
- Published
- 2018
85. Recursive formula for the double-barrier Parisian stopping time
- Author
-
Jia Wei Lim and Angelos Dassios
- Subjects
Statistics and Probability ,050208 finance ,Recursion ,Parisian stopping times ,Laplace transform ,General Mathematics ,05 social sciences ,Double-sided Parisian options ,Brownian excursion ,Double barrier ,01 natural sciences ,Brownian excursions ,Formal proof ,010104 statistics & probability ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Stopping time ,0502 economics and business ,Probabilistic proof ,Calculus ,Applied mathematics ,QA Mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we obtain a recursive formula for the density of the double-barrier Parisian stopping time. We present a probabilistic proof of the formula for the first few steps of the recursion, and then a formal proof using explicit Laplace inversions. These results provide an efficient computational method for pricing double-barrier Parisian options.
- Published
- 2018
86. On the Parisian ruin of the dual Lévy risk model
- Author
-
Chen Yang, Zhong Li, and Kristian P. Sendova
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,05 social sciences ,Function (mathematics) ,Poisson distribution ,01 natural sciences ,Lévy process ,Dual (category theory) ,010104 statistics & probability ,symbols.namesake ,Risk model ,0502 economics and business ,symbols ,Penalty method ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematical economics ,Mathematics - Abstract
In this paper we investigate the Parisian ruin problem of the general dual Lévy risk model. Unlike the usual concept of ultimate ruin, allowing the surplus level to be negative within a prespecified period indicates that the deficit at Parisian ruin is not necessarily equal to zero. Hence, we consider a Gerber–Shiu type expected discounted penalty function at the Parisian ruin and obtain an explicit expression for this function under the dual Lévy risk model. As particular cases, we calculate the Parisian ruin probability and the expected discountedkth moments of the deficit at the Parisian ruin for the compound Poisson dual risk model and a drift-diffusion model. Numerical examples are given to illustrate the behavior of Parisian ruin and the expected discounted deficit at Parisian ruin.
- Published
- 2017
87. The failure probability of components in three-state networks with applications to age replacement policy
- Author
-
Somayeh Ashrafi and Majid Asadi
- Subjects
Statistics and Probability ,021103 operations research ,Signature matrix ,General Mathematics ,Failure probability ,0211 other engineering and technologies ,02 engineering and technology ,State (functional analysis) ,01 natural sciences ,Stochastic ordering ,Signature (logic) ,010104 statistics & probability ,0101 mathematics ,Statistics, Probability and Uncertainty ,Algorithm ,Mathematics - Abstract
In this paper we investigate the stochastic properties of the number of failed components of a three-state network. We consider a network made up of n components which is designed for a specific purpose according to the performance of its components. The network starts operating at time t = 0 and it is assumed that, at any time t > 0, it can be in one of states up, partial performance, or down. We further suppose that the state of the network is inspected at two time instants t1 and t2 (t1 < t2). Using the notion of the two-dimensional signature, the probability of the number of failed components of the network is calculated, at t1 and t2, under several scenarios about the states of the network. Stochastic and ageing properties of the proposed failure probabilities are studied under different conditions. We present some optimal age replacement policies to show applications of the proposed criteria. Several illustrative examples are also provided.
- Published
- 2017
88. Limit laws on extremes of nonhomogeneous Gaussian random fields
- Author
-
Zhongquan Tan
- Subjects
Statistics and Probability ,Limit of a function ,Statistics::Theory ,Random field ,Fractional Brownian motion ,General Mathematics ,Gaussian ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics::Probability ,Gumbel distribution ,symbols ,Limit (mathematics) ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper, by using the exact tail asymptotics derived by Debicki, Hashorva and Ji (Ann. Probab. 2014), we proved the Gumbel limit theorem for the maximum of a class of non-homogeneous Gaussian random fields. By using the obtained results, we also derived the Gumbel laws for Shepp statistics of fractional Brownian motion and Gaussian integrated process as well as the Gumbel law for Storage process with fractional Brownian motion as
- Published
- 2017
89. Stochastic comparison of parallel systems with heterogeneous exponential components
- Author
-
Jiantian Wang and Bin Cheng
- Subjects
Statistics and Probability ,010104 statistics & probability ,General Mathematics ,010102 general mathematics ,Order statistic ,Parallel computing ,0101 mathematics ,Statistics, Probability and Uncertainty ,Residual ,01 natural sciences ,Mathematics ,Exponential function - Abstract
In this paper we provide a sufficient condition for mean residual life ordering of parallel systems with n ≥ 3 heterogeneous exponential components.
- Published
- 2017
90. Increasing convex order on generalized aggregation of SAI random variables with applications
- Author
-
Xiaoqing Pan and Xiaohu Li
- Subjects
Statistics and Probability ,050208 finance ,General Mathematics ,Reliability (computer networking) ,05 social sciences ,Regular polygon ,Monotonic function ,01 natural sciences ,Convexity ,Submodular set function ,010104 statistics & probability ,0502 economics and business ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Linear combination ,Majorization ,Random variable ,Mathematics - Abstract
In this paper we study general aggregation of stochastic arrangement increasing random variables, including both the generalized linear combination and the standard aggregation as special cases. In terms of monotonicity, supermodularity, and convexity of the kernel function, we develop several sufficient conditions for the increasing convex order on the generalized aggregations. Some applications in reliability and risks are also presented.
- Published
- 2017
91. Rare events of transitory queues
- Author
-
Harsha Honnappa
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Queueing theory ,021103 operations research ,General Mathematics ,Probability (math.PR) ,0211 other engineering and technologies ,Workload ,02 engineering and technology ,01 natural sciences ,Computer Science::Performance ,010104 statistics & probability ,FOS: Mathematics ,Rare events ,Applied mathematics ,Large deviations theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Random variable ,Mathematics - Probability ,Empirical process ,Mathematics - Abstract
We study the rare-event behavior of the workload process in a transitory queue, where the arrival epochs (or 'points') of a finite number of jobs are assumed to be the ordered statistics of independent and identically distributed (i.i.d.) random variables. The service times (or 'marks') of the jobs are assumed to be i.i.d. random variables with a general distribution, that are jointly independent of the arrival epochs. Under the assumption that the service times are strictly positive, we derive the large deviations principle (LDP) satisfied by the workload process. The analysis leverages the connection between ordered statistics and self-normalized sums of exponential random variables to establish the LDP. In this paper we present the first analysis of rare events in transitory queueing models, supplementing prior work that has focused on fluid and diffusion approximations.
- Published
- 2017
92. Almost stochastic dominance under inconsistent utility and loss functions
- Author
-
Chin Hon Tan, Zhou He, and Chunling Luo
- Subjects
Statistics and Probability ,Current (mathematics) ,General Mathematics ,Stochastic dominance ,010501 environmental sciences ,01 natural sciences ,010104 statistics & probability ,Econometrics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Set (psychology) ,Decision model ,Random variable ,0105 earth and related environmental sciences ,Mathematics - Abstract
Current literature on stochastic dominance assumes utility/loss functions to be the same across random variables. However, decision models with inconsistent utility functions have been proposed in the literature. The use of inconsistent loss functions when comparing between two random variables can also be appropriate under other problem settings. In this paper we generalize almost stochastic dominance to problems with inconsistent utility/loss functions. In particular, we propose a set of conditions that is necessary and sufficient for clear preferences when the utility/loss functions are allowed to vary across different random variables.
- Published
- 2017
93. Absolute continuity of distributions of one-dimensional Lévy processes
- Author
-
Tongkeun Chang
- Subjects
Statistics and Probability ,Pure mathematics ,Logarithm ,Subordinator ,General Mathematics ,010102 general mathematics ,Absolute continuity ,Lebesgue integration ,01 natural sciences ,Lévy process ,010104 statistics & probability ,symbols.namesake ,symbols ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In this paper we study the existence of Lebesgue densities of one-dimensional Lévy processes. Equivalently, we show the absolute continuity of the distributions of one-dimensional Lévy processes. Compared with the previous literature, we consider Lévy processes with Lévy symbols of a logarithmic behavior at ∞.
- Published
- 2017
94. Optimal bulking threshold of batch service queues
- Author
-
Yun Zeng and Cathy H. Xia
- Subjects
Statistics and Probability ,Service (business) ,Queueing theory ,Mathematical optimization ,Schedule ,021103 operations research ,General Mathematics ,Quality of service ,0211 other engineering and technologies ,02 engineering and technology ,01 natural sciences ,Telecommunications network ,010104 statistics & probability ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Time complexity ,Mathematics - Abstract
Batch service has a wide application in manufacturing, communication networks, and cloud computing. In batch service queues with limited resources, one critical issue is to properly schedule the service so as to ensure the quality of service. In this paper we consider an M/G[a,b]/1/N batch service queue with bulking threshold a, max service capacity b, and buffer capacity N, where N can be finite or infinite. Through renewal theory, busy period analysis and decomposition techniques, we demonstrate explicitly how the bulking threshold influences the system performance such as the mean waiting time and time-averaged number of loss customers in batch service queues. We then establish a necessary and sufficient condition on the optimal bulking threshold that minimizes the expected waiting time. Enabled by this condition, we propose a simple algorithm which guarantees to find the optimal threshold in polynomial time. The performance of the algorithm is also demonstrated by numerical examples.
- Published
- 2017
95. Large deviations for the stochastic predator–prey model with nonlinear functional response
- Author
-
Krishnan Balachandran and M. Suvinthra
- Subjects
Statistics and Probability ,Weak convergence ,General Mathematics ,Gaussian ,010102 general mathematics ,Functional response ,01 natural sciences ,Stochastic partial differential equation ,010104 statistics & probability ,Nonlinear system ,symbols.namesake ,symbols ,Large deviations theory ,Statistical physics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Rate function ,Randomness ,Mathematics - Abstract
In this paper we consider a diffusive stochastic predator–prey model with a nonlinear functional response and the randomness is assumed to be of Gaussian nature. A large deviation principle is established for solution processes of the considered model by implementing the weak convergence technique.
- Published
- 2017
96. A unified approach for drawdown (drawup) of time-homogeneous Markov processes
- Author
-
Hongzhong Zhang, Bin Li, and David Landriault
- Subjects
Statistics and Probability ,050208 finance ,Stochastic process ,General Mathematics ,05 social sciences ,Markov process ,Mathematical Finance (q-fin.MF) ,01 natural sciences ,Integral equation ,Lévy process ,Infimum and supremum ,FOS: Economics and business ,010104 statistics & probability ,symbols.namesake ,Quantitative Finance - Mathematical Finance ,0502 economics and business ,60G07, 60G40 ,Drawdown (hydrology) ,Overshoot (signal) ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,First-hitting-time model ,Mathematics - Abstract
Drawdown (resp. drawup) of a stochastic process, also referred as the reflected process at its supremum (resp. infimum), has wide applications in many areas including financial risk management, actuarial mathematics and statistics. In this paper, for general time-homogeneous Markov processes, we study the joint law of the first passage time of the drawdown (resp. drawup) process, its overshoot, and the maximum of the underlying process at this first passage time. By using short-time pathwise analysis, under some mild regularity conditions, the joint law of the three drawdown quantities is shown to be the unique solution to an integral equation which is expressed in terms of fundamental two-sided exit quantities of the underlying process. Explicit forms for this joint law are found when the Markov process has only one-sided jumps or is a L\'{e}vy process (possibly with two-sided jumps). The proposed methodology provides a unified approach to study various drawdown quantities for the general class of time-homogeneous Markov processes., Comment: 24 pages, 2 figures
- Published
- 2017
97. Moderate deviation principles for importance sampling estimators of risk measures
- Author
-
Pierre Nyquist
- Subjects
Statistics and Probability ,Deviation risk measure ,021103 operations research ,General Mathematics ,Risk measure ,0211 other engineering and technologies ,Estimator ,02 engineering and technology ,01 natural sciences ,010104 statistics & probability ,Expected shortfall ,Statistics ,Econometrics ,Large deviations theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Value at risk ,Importance sampling ,Quantile ,Mathematics - Abstract
Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.
- Published
- 2017
98. Statistical inference for partially observed branching processes with immigration
- Author
-
I. Rahimov
- Subjects
Statistics and Probability ,education.field_of_study ,Stationary distribution ,Offspring ,General Mathematics ,010102 general mathematics ,Population ,Estimator ,01 natural sciences ,Branching (linguistics) ,010104 statistics & probability ,Statistics ,Statistical inference ,Quantitative Biology::Populations and Evolution ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Branching process ,Mathematics - Abstract
In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.
- Published
- 2017
99. Asymptotic results for the multiple scan statistic
- Author
-
Michael V. Boutsikas, Fotios S. Milienos, and Markos V. Koutras
- Subjects
Statistics and Probability ,Sequence ,050208 finance ,Scan statistic ,General Mathematics ,05 social sciences ,Random walk ,Poisson distribution ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Total variation ,Convergence of random variables ,0502 economics and business ,Statistics ,symbols ,Bernoulli trial ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Statistic ,Mathematics - Abstract
The contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.
- Published
- 2017
100. Epidemic risk and insurance coverage
- Author
-
Matthieu Simon, Claude Lefèvre, Philippe Picard, Université libre de Bruxelles (ULB), Laboratoire de Sciences Actuarielle et Financière (SAF), Université Claude Bernard Lyon 1 (UCBL), and Université de Lyon-Université de Lyon
- Subjects
Statistics and Probability ,education.field_of_study ,050208 finance ,Actuarial science ,General Mathematics ,05 social sciences ,Population ,01 natural sciences ,010104 statistics & probability ,Insurance premium ,0502 economics and business ,Epidemic outbreak ,[MATH]Mathematics [math] ,0101 mathematics ,Statistics, Probability and Uncertainty ,Martingale (probability theory) ,Epidemic model ,education ,ComputingMilieux_MISCELLANEOUS ,health care economics and organizations ,Insurance coverage ,Mathematics - Abstract
In this paper we aim to apply simple actuarial methods to build an insurance plan protecting against an epidemic risk in a population. The studied model is an extended SIR epidemic in which the removal and infection rates may depend on the number of registered removals. The costs due to the epidemic are measured through the expected epidemic size and infectivity time. The premiums received during the epidemic outbreak are measured through the expected susceptibility time. Using martingale arguments, a method by recursion is developed to calculate the cost components and the corresponding premium levels in this extended epidemic model. Some numerical examples illustrate the effect of removals and the premium calculation in an insurance plan.
- Published
- 2017
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