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We examine the influence of using a restart mechanism on the stationary distributions of a particular class of Markov chains. Namely, we consider a family of multivariate autoregressive stochastic sequences that restart when hit a neighbourhood of the origin, and study their distributional limits when the autoregressive coefficient tends to one, the noise scaling parameter tends to zero, and the neighbourhood size varies. We show that the restart mechanism may change significantly the limiting distribution. We obtain a limit theorem with a novel type of limiting distribution, a mixture of an atomic distribution and an absolutely continuous distribution whose marginals, in turn, are mixtures of distributions of signed absolute values of normal random variables. In particular, we provide conditions for the limiting distribution to be normal, like in the case without restart mechanism. The main theorem is accompanied by a number of examples and auxiliary results of their own interest.

In this paper, we derive a higher-order asymptotic expansion of characteristic functions of an Ito semimartingale over asymptotically shrinking time intervals. The leading term in the expansion is determined by the value of the diffusive coefficient at the beginning of the interval. The higher-order terms are determined by the jump compensator as well as the coefficients appearing in the diffusion dynamics. The result is applied to develop a nearly rate-efficient estimator of the leverage coefficient of an asset price, i.e., the coefficient in its volatility dynamics that appears in front of the Brownian motion that drives also the asset price.

Let $$X_1,\ldots ,X_n$$ be an i.i.d. sample from symmetric stable distribution with stability parameter $$\alpha$$ and scale parameter $$\gamma$$ . Let $$\varphi _n$$ be the empirical characteristic function. We prove a uniform large deviation inequality: given preciseness $$\epsilon >0$$ and probability $$p\in (0,1)$$ , there exists universal (depending on $$\epsilon$$ and p but not depending on $$\alpha$$ and $$\gamma$$ ) constant $$\bar{r}>0$$ so that $$P\big (\sup _{u>0:r(u)\le \bar{r}}|r(u)-\hat{r}(u)|\ge \epsilon \big )\le p,$$ where $$r(u)=(u\gamma )^{\alpha }$$ and $$\hat{r}(u)=-\ln |\varphi _n(u)|$$ . As an applications of the result, we show how it can be used in estimation the unknown stability parameter $$\alpha$$ .

The aim of the present work is evaluation of absolute constants in the Arak inequalities for the concentration functions of convolutions of probability distributions. This result allows us to calculate the constant in the inequality for the uniform distance between n and (n + 1)-fold convolutions of one-dimensional symmetric probability distributions with a characteristic function separated from −1, as well as a number of other estimates, in particular, the accuracy of approximation of samples of rare events by the Poisson point process.

In this paper, we study the finite-time ruin probability in the risk model driven by a Levy subordinator, by incorporating the popular Fourier-cosine method. Our interest is to propose a general approximation for any specified precision provided that the characteristic function of the Levy Process is known. To achieve this, we derive an explicit integral expression for the finite-time ruin probability, which is expressed in terms of the density function and the survival function of L t . Moreover, we apply the rearrangement inequality to further improve our approximations. In addition, with only mild and practically relevant assumptions, we prove that the approximation error can be made arbitrarily small (actually an algebraic convergence rate up to 3, which is the fastest possible approximant known upon all in the literature), and has a linear computation complexity in a number of terms of the Fourier-cosine expansion. The effectiveness of our results is demonstrated in various numerical studies; through these examples, the supreme power of the Fourier-cosine method is once demonstrated.

We propose generalization of the spatial Simultaneous Autoregressive (SAR) model on a lattice towards modelling for asymmetry. Under the assumption of skew-normal error structure, expression for density and characteristic function for the induced distribution of response are obtained. Full-likelihood based implementation of the proposed model to a real data set is performed using Differential Evolution (DE). The relevant results are reported and compared with the results from existing models.

We propose an extension of the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model taking into account jump clustering phenomena. We assume that the intensity process of the Hawkes driver coincides, up to a constant, with the variance process. By applying the theory of continuous-state branching processes with immigration, we prove existence and uniqueness of strong solutions of the SDE governing the asset price dynamics. We propose a measure change of self-exciting Esscher type in order to describe the relation between the risk-neutral and the historical dynamics, showing that the $$\Gamma $$ Γ -OU Hawkes framework is stable under this probability change. By exploiting the affine features of the model we provide an explicit form for the Laplace transform of the asset log-return, for its quadratic variation and for the ergodic distribution of the variance process. We show that the proposed model exhibits a larger flexibility in comparison with the $$\Gamma $$ Γ -OU model, in spite of the same number of parameters required. We calibrate the model on market vanilla option prices via characteristic function inversion techniques, we study the price sensitivities and propose an exact simulation scheme. The main financial achievement is that implied volatility of options written on VIX is upward shaped due to the self-exciting property of Hawkes processes, in contrast with the usual downward slope exhibited by the $$\Gamma $$ Γ -OU Barndorff-Nielsen and Shephard model.

The concept of extended slash distribution is introduced. For the extended slash distribution of the sum of two independent logistic random variables, the characteristic function, moments, absolute...

Here we consider a generalized log-transformed version of the Weibull distribution and investigate some of its important properties like expressions for the cumulative distribution function hazard rate function, quantile function, characteristic function, raw moments, incomplete moments, etc. The distribution and moments of order statistics are obtained along with some results on certain structural properties of the distribution. The maximum likelihood estimation of the parameters of the distribution is attempted for both complete and censored data sets and the usefulness of the distribution is illustrated with the help of real-life data sets from biomedical fields.

A matrix variate generalization of the two-sided power distribution is proposed. Several properties of this distribution including probability density function, cumulative distribution function and characteristic function are derived.

Several researchers have addressed the problem of testing the homogeneity of the scale parameters of several independent inverse Gaussian distributions based on the likelihood ratio test. However, only approximations of the distribution function of the test statistic are available in the literature. In this note, we present the exact distribution of the likelihood ratio test statistic for testing the equality of the scale parameters of several independent inverse Gaussian populations in a closed form. To this end, we apply the Mellin inverse transform and the Jacobi polynomial expansion to the moments of the likelihood ratio test statistic. We also propose an approximate method based on the Jacobi polynomial expansion. Finally, we apply an accurate numerical method, which is based on the inverse of characteristic function, to obtain a near-exact approximation of the likelihood ratio test statistic distribution. The proposed methods are illustrated via numerical and real data examples.

Research, development and extensive use of generalized form of distributions in order to analyze and modeling of applied sciences research data has been growing tremendously. Weibull and Frechet distribution are widely discussed for reliability and survival analysis using experimental data from physical, chemical, environmental and engineering sciences. Both the distributions are applicable to extreme value theory as well as small and large data sets. Recently researchers develop several probability distributions to model experimental data as these parent models are not adequate to fit in some experiments. Modified forms of the Weibull distribution and Frechet distribution are more flexible distributions for modeling experimental data. This article aims to introduce a generalize form of Weibull distribution known as Frechet-Weibull Distribution (FWD) by using the T-X family which extends a more flexible distribution for modeling experimental data. Here the pdf and cdf with survival function [S(t)], hazard rate function [h(t)] and asymptotic behaviour of pdf and survival function and the possible shapes of pdf, cdf, S(t) and h(t) of FWD have been studied and the parameters are estimated using maximum livelihood method (MLM). Some statistical properties of FWD such as mode, moments, skewness, kurtosis, variation, quantile function, moment generating function, characteristic function and entropies are investigated. Finally the FWD has been applied to two sets of observations from mechanical engineering and shows the superiority of FWD over other related distributions. This study will provide a useful tool to analyze and modeling of datasets in Mechanical Engineering sciences and other related field.

This paper is devoted to a nonparametric estimation of the probability θ : = ℙ ( X < Y ) , where X, Y are continuous univariate random variables of interest and observed with additional random erro...

This paper deals with testing for nondegenerate normality of a $d$-variate random vector $X$ based on a random sample $X_1,\ldots,X_n$ of $X$. The rationale of the test is that the characteristic function $\psi(t) = \exp(-\|t\|^2/2)$ of the standard normal distribution in $\mathbb{R}^d$ is the only solution of the partial differential equation $\Delta f(t) = (\|t\|^2-d)f(t)$, $t \in \mathbb{R}^d$, subject to the condition $f(0) = 1$. By contrast with a recent approach that bases a test for multivariate normality on the difference $\Delta \psi_n(t)-(\|t\|^2-d)\psi(t)$, where $\psi_n(t)$ is the empirical characteristic function of suitably scaled residuals of $X_1,\ldots,X_n$, we consider a weighted $L^2$-statistic that employs $\Delta \psi_n(t)-(\|t\|^2-d)\psi_n(t)$. We derive asymptotic properties of the test under the null hypothesis and alternatives. The test is affine invariant and consistent against general alternatives, and it exhibits high power when compared with prominent competitors., Comment: 16 pages, 1 figure, 6 tables

In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a main concern for researchers. A cumulative distribution function (CDF) of these random variables (Fi:n) is a complex manipulating, long time consuming and a software-intensive tool that takes more and more times. Therefore, obtaining approximations and boundaries for Fi:n and other theoretical properties of these variables, such as moments, quantiles, characteristic function, and some related probabilities, has always been a main chal- lenge. Recently, Bayramoglu (2018) provided a new definition of ordering, by point to point ordering Fi’s (D-order) and showed that these new functions are CDFs and also, the corresponding random variables are independent. Thus, he suggested new CDFs (F[i]) that can be used as an alternative of Fi:n. Now with using, just F[1], and F[n], we have found the upper and lower bounds of Fi:n. Furthermore, specially a precisely approximation for F1:n and Fn:n (F1;n:n). Also in many cases approximations for other CDFs are derived. In addition, we compare approximated function with those oered by Bayramoglu and it is shown that our results of these proposed functions are far better than D-order functions.

This article studies the properties of the characteristic function of fractional stable distribution expressed through the Mittag-Leffler function. It is shown that the existing integral expression for the Mittag-Leffler function is incorrect, and the corrected integral expression is given. New properties of the characteristic function of the fractional stable law are obtained that make it possible to perform the inverse Fourier transformation. As a result, the integral representations are obtained for the density and distribution function of the fractional stable law. Some properties of these representations are studied, and the results of numerical calculations for the probability density and distribution function are presented.

We deal with the fractional Ornstein–Uhlenbeck (fO–U) process driven by the fractional Brownian motion (fBm), where the drift parameter $$\alpha $$ of the fO–U process is any unknown real number, whereas the Hurst index H of the fBm belongs to (0, 1) and is assumed to be known. Under this setting we consider the least squares (LS)-based estimators and the maximum likelihood estimator (MLE) of $$\alpha $$ , and examine the efficiencies of the LS-based estimators relative to the MLE, paying attention to the effect of the sign of $$\alpha $$ and the value of H. It is found that the MLE is more efficient than the LSE when $$\alpha \ne 0$$ , but the LSE is more efficient when $$\alpha =0$$ .

The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat Hsu’s (J. Chinese Math. Soc. N.S. 1, 257–280, 1951) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace–Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly Hall’s (Z. Wahrsch. Verw. Gebiete 62, 355–359, 1983) result for distributions on the right half-line.

A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank–based distance correlation computed over two complementary order-preserved sets. General properties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth–order degenerate V -statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.

This paper introduces a novel model with six parameters called “flexible additive Weibull distribution (FAWD).” The suggested model is capable to model the life time data with a non-monotonic hazar...

The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0

In this paper, a univariate finite mixed generalized normal distribution (MixGND) is proposed. First, we derive some probabilistic properties including hazard rate function, characteristic function...

The aim of this paper is to derive a set of easily implementable formulas regarding the probability distribution of the integral of a squared Brownian motion with drift. By reestablishing the characteristic function via the Karhunen-Loeve transform, we obtain recurrence formulas for the moments as well as rapidly converging series with explicit coefficients for the probability density function and cumulative distribution function. We also perform asymptotic analyses to obtain sharp approximations for the exact formulas with small or large arguments. Extensive use is made of the parabolic cylinder function. Numerical experiments are conducted to demonstrate the applicability and efficiency of the proposed formulas as well as how they vary under the drift impact.

By introducing the notion of “empirical likelihood function of observing sums”, unequally-spaced time series data and longitudinal data generated from Levy processes can be analyzed. Characteristic function is further incorporated to handle the situations where the density function of the increments is difficult to obtain. In the situations where both characteristic function and the density function are available, it is shown through the simulation examples that the proposed empirical maximum likelihood method does not suffer from significant information loss comparing to the maximum likelihood estimation method. The performances of the proposed method is tested for both equally-spaced and unequally-spaced observations.

We introduce the notion of a regime switching affine process. Informally this is a Markov process that behaves conditionally on each regime as an affine process with specific parameters. To facilitate our analysis, specific restrictions are imposed on these parameters. The regime switches are driven by a Markov chain. We prove that the joint process of the Markov chain and the conditionally affine part is a process with an affine structure on an enlarged state space, conditionally on the starting state of the Markov chain. Like for affine processes, the characteristic function can be expressed in a set of ordinary differential equations that can sometimes be solved analytically. This result unifies several semi-analytical solutions found in the literature for pricing derivatives of specific regime switching processes on smaller state spaces. It also provides a unifying theory that allows us to introduce regime switching to the pricing of many derivatives within the broad class of affine processes. Examples include European options and term structure derivatives with stochastic volatility and default. Essentially, whenever there is a pricing solution based on an affine process, we can extend this to a regime switching affine process without sacrificing the analytical tractability of the affine process.

Motivated by a (seemingly previously unnoticed) result stating that d −dimensional distributions on $(0,\infty )^{d}$ are characterized by the collection of their min-linear projections, we introduce and study a notion of min-characteristic function (min-CF) of a random vector with strictly positive components. Unlike the related notion of max-characteristic function which has been studied recently, the existence of the min-CF does not hinge on any integrability conditions. It is itself a multivariate distribution function, which is continuous and concave, no matter which properties the initial distribution function has. We show the equivalence between convergence in distribution and pointwise convergence of min-CFs, and we also study the functional convergence of the min-CF of the empirical distribution function of a sample of independent and identically distributed random vectors. We provide some further insight into the structure of the set of min-CFs, and we conclude by showing how transforming the components of an arbitrary random vector by a suitable one-to-one transformation such as the exponential function allows the construction of a notion of min-CF for arbitrary random vectors.

We propose three new characterizations and corresponding distance-based weighted test criteria for the two-sample problem, and for testing symmetry and independence with multivariate data. All quantities have the common feature of involving characteristic functions, and it is seen that these quantities are intimately related to some earlier methods, thereby generalizing them. The connection rests on a special choice of the weight function involved. Equivalent expressions of the distances in terms of densities are given as well as a Bayesian interpretation of the weight function is involved. The asymptotic behavior of the tests is investigated both under the null hypothesis and under alternatives, and affine invariant versions of the test criteria are suggested. Numerical studies are conducted to examine the performances of the criteria. It is shown that the normal weight function, which is the hitherto most often used, is seriously suboptimal. The procedures are biased in the sense that the corresponding test criteria degenerate in high dimension and hence a bias correction is required as the dimension tends to infinity.

In this paper, we propose a kernel-type estimator for the local characteristic function of locally stationary processes. Under weak moment conditions, we prove joint asymptotic normality for local empirical characteristic functions. For time-varying linear processes, we establish a central limit theorem under the assumption of finite absolute first moments of the process. Additionally, we prove weak convergence of the local empirical characteristic process. We apply our asymptotic results to parameter estimation. Furthermore, by extending the notion of distance correlation of Szekely, Rizzo and Bakirov (2007) to locally stationary processes, we are able to provide asymptotic theory for local empirical distance correlations. Finally, we provide a simulation study on minimum distance estimation for a-stable distributions and illustrate the pairwise dependence structure over time of log returns of German stock prices via local empirical distance correlations.

In this paper the authors study the problem of testing the hypothesis of a doubly exchangeable covariance matrix for three-level multivariate observations, taken on m variables over u sites and over v time/space points. Through the decomposition of the main hypothesis into a set of three sub-hypotheses, the likelihood ratio test statistic is defined, its exact moments are determined, and its exact distribution is studied. Because this distribution is very much intricate, a very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this near-exact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes. A simulation study is also conducted and two real-data examples are presented.

We propose a novel class of independence measures for testing independence between two random vectors based on the discrepancy between the conditional and the marginal characteristic functions. The...

In this study, we present a multivariate test for population bioequivalence. Recently, Chervoneva et al. (2007) proposed a multivariate generalization of the criteria for testing univariate population bioequivalence by obtaining an approximate test based on an unbiased estimator of linearized criteria. Their test involved decomposing the unbiased estimator into several terms and using the existing interval estimates of these terms. Although this test’s type I error probability is often lower than the nominal significance level, its approximation accuracy is not good in case of small samples. This study intends to present a more accurate approximate test. To improve the accuracy, it is important to investigate the distribution of the unbiased estimator of the linearized bioequivalence criteria. Therefore, we derive an explicit representation of its characteristic function and asymptotic distribution and propose new tests that exploit the asymptotic results. We additionally obtain the theoretical asymptotic power function in an explicit manner and investigate the local powers of our test, proving that they are asymptotically unbiased. Further, we investigate their finite-sample performance via Monte Carlo simulations and confirm that the accuracy is better as compared to that observed in a previous test. Finally, we use our methods to analyze real data.

In this article, we compute and estimate the stress-strength reliability R=P(A′X + B′Y + C > 0) when X and Y are distributed independently as multivariate skewed generalized Pearson type II (SGPII)...

Irrational-behavior-proof (IBP) conditions are important aspects to keep stable cooperation in dynamic cooperative games. In this paper, we focus on the establishment of IBP conditions. Firstly, the relations of three kinds of IBP conditions are described. An example is given to show that they may not hold, which could lead to the fail of cooperation. Then, based on a kind of limit characteristic function, all these conditions are proved to be true along the cooperative trajectory in a transformed cooperative game. It is surprising that these facts depend only upon the individual rationalities of players for the Shapley value and the group rationalities of players for the core. Finally, an illustrative example is given.

According to the Wiener–Hopf factorization, the characteristic function $$\varphi $$ of any probability distribution $$\mu $$ on $$\mathbb {R}$$ can be decomposed in a unique way as $$\begin{aligned} 1-s\varphi (t)=[1-\chi _-(s,it)][1-\chi _+(s,it)],\quad |s|\le 1,\,t\in \mathbb {R}\,, \end{aligned}$$where $$\chi _-(e^{iu},it)$$ and $$\chi _+(e^{iu},it)$$ are the characteristic functions of possibly defective distributions in $$\mathbb {Z}_+\times (-\infty ,0)$$ and $$\mathbb {Z}_+\times [0,\infty )$$, respectively. We prove that $$\mu $$ can be characterized by the sole data of the upward factor $$\chi _+(s,it)$$, $$s\in [0,1)$$, $$t\in \mathbb {R}$$ in many cases including the cases where: We conjecture that any probability distribution is actually characterized by its upward factor. This conjecture is equivalent to the following: Any probability measure$$\mu $$on$$\mathbb {R}$$whose support is not included in$$(-\,\infty ,0)$$is determined by its convolution powers$$\mu ^{*n}$$, $$n\ge 1$$restricted to$$[0,\infty )$$. We show that in many instances, the sole knowledge of $$\mu $$ and $$\mu ^{*2}$$ restricted to $$[0,\infty )$$ is actually sufficient to determine $$\mu $$. Then we investigate the analogous problem in the framework of infinitely divisible distributions.

We give sufficient identifiability conditions for estimating mixing proportions in two-component mixtures of skew normal distributions with one known component. We consider the univariate case as well as two multivariate extensions: a multivariate skew normal distribution (MSN) by Azzalini and Dalla Valle (1996) and the canonical fundamental skew normal distribution (CFUSN) by Arellano-Valle and Genton (2005). The characteristic function of the CFUSN distribution is additionally derived.

Thermodynamic quantities, like heat, entropy, or work, are random variables, in stochastic systems. Here, we investigate the statistics of the heat exchanged by a Brownian particle subjected to a logarithm-harmonic potential. We derive analytically the characteristic function and its moments for the heat. Through numerical integration, and numerical simulation, we calculate the probability distribution as well, characterizing fully the statistical behavior of the heat. The results are also investigated in the asymptotic limit, where we encounter the characteristic function in terms of hypergeometric functions., 8 pages, 2 figures

The procedures to discover proper new models in probability theory for different data collections are highly prevalent these days among the researchers of this area whenever existing literature models are not appropriate. Before delivering a product, manufacturers of raw materials or finished materials must follow some compliance standards in various engineering disciplines to avoid severe losses. Materials of high strength are necessary to ensure the safety of human lives along with infrastructures to elude the significant obligations linked with the provisions of non-compliant products. Using probability theory, we introduce the weighted version of inverted Kumaraswamy Distribution, which could be considered a better model than some other sub-models used to model Carbon fiber’s strength data. We derive various statistical properties of this distribution such as cumulative distribution, moments, mean residual life, reversed residual life functions, moment generating function, characteristic function, harmonic mean, and geometric mean. Parameters are estimated through the maximum likelihood method and ordinary moments. Simulation studies are carried out to illustrate the theoretical results of these two approaches. Furthermore, two real data sets of Carbon fibers strength are utilized to contrast the proposed model and its sub-models like inverted Kumaraswamy distribution and Kumaraswamy Sushila distribution through different goodness of fit criteria such as Akaike Information Criterion (AIC), corrected Akaike information criterion, and the Bayesian Information Criterion (BIC). Results reveal the outperformance of the proposed model compared to other models, which render it a proper interchange of the current sub-models.

We propose an estimator for the mean of random variables in separable real Banach spaces using the empirical characteristic function. Assuming that the covariance operator of the random variable is bounded in a precise sense, we show that the proposed estimator achieves the optimal sub-Gaussian rate, except for a faster decaying mean-dependent term. Under a mild condition, an iterative refinement procedure can essentially eliminate the mean-dependent term and provide a shift-equivariant estimate. Our results particularly suggests that a certain Gaussian width that appears in the best known rate in the literature might not be necessary. Furthermore, using the boundedness of the characteristic functions, we also show that, except possibly at high signal-to-noise ratios, the proposed estimator is gracefully robust against adversarial "contamination". Our analysis is overall concise and transparent, thanks to the tractability of the characteristic functions., Comment: some minor corrections

We consider model-free monitoring procedures for strict stationarity of a given time series. The new criteria are formulated as L2-type statistics incorporating the multivariate empirical characteristic function. Asymptotic results are obtained for the closed-end scenario and Monte Carlo results are presented. The new methods are also employed in order to test for possible stationarity breaks in time-series data from the financial sector.

A probability distribution μ on R d is quasi-infinitely divisible if its characteristic function has the representation μ = μ 1 ∕ μ 2 with infinitely divisible distributions μ 1 and μ 2 . In Lindner et al. (2018, Thm. 4.1) it was shown that the class of quasi-infinitely divisible distributions on R is dense in the class of distributions on R with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on R d is not dense in the class of distributions on R d with respect to weak convergence if d ≥ 2 .

We study a novel class of affine invariant and consistent tests for normality in any dimension. The tests are based on a characterization of the standard $d$-variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schr\"odinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood-of-model validation procedure., Comment: 29 pages, 1 figure, 7 tables

It is a typical standard assumption in the density deconvolution problem that the characteristic function of the measurement error distribution is non-zero on the real line. While this condition is assumed in the majority of existing works on the topic, there are many problem instances of interest where it is violated. In this paper we focus on non--standard settings where the characteristic function of the measurement errors has zeros, and study how zeros multiplicity affects the estimation accuracy. For a prototypical problem of this type we demonstrate that the best achievable estimation accuracy is determined by the multiplicity of zeros, the rate of decay of the error characteristic function, as well as by the smoothness and the tail behavior of the estimated density. We derive lower bounds on the minimax risk and develop optimal in the minimax sense estimators. In addition, we consider the problem of adaptive estimation and propose a data-driven estimator that automatically adapts to unknown smoothness and tail behavior of the density to be estimated., 32 pages

We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK–V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting.

We develop a nonparametric test for deciding whether a semimartingale process, modeling an asset price, contains a fixed time of discontinuity, i.e., a positive probability of a jump, at a given point in time, and we further propose a rate-optimal estimator of the jump distribution when this is the case. Ito semimartingales used commonly in applied work have absolutely continuous in time, with respect to Lebesgue measure, jump compensators, and this rules out fixed times of discontinuity in their paths. However, certain phenomena, such as scheduled economic announcements in finance, make the existence of such discontinuities a possibility. The inference in the paper is based on noisy observations of options written on the asset with different strikes and two different expiration dates. The asymptotics is joint in which the times to maturity of the options shrink to zero and the number of observed options increases to infinity. The test is based on estimates of the characteristic function of the increments of the semimartingale, constructed from the option data, and the fact that the asymptotic limit of the increments and their characteristic functions is different with and without fixed time of discontinuity. The limit distribution of the test statistic is derived and feasible inference is developed on the basis of wild bootstrap type techniques. A Monte Carlo and an empirical illustration show the applicability of the developed inference procedures.

We investigate long and short memory in α‐stable moving averages and max‐stable processes with α‐Fréchet marginal distributions. As these processes are heavy‐tailed, we rely on the notion of long range dependence based on the covariance of indicators of excursion sets. Sufficient conditions for the long and short range dependence of α‐stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max‐stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence., publishedVersion

Using the spectral theorem we compute the Quantum Fourier Transform (or Vacuum Characteristic Function) $\langle \Phi, e^{itH}\Phi\rangle$ of an observable $H$ defined as a self-adjoint sum of the generators of a finite-dimensional Lie algebra, where $\Phi$ is a unit vector in a Hilbert space $\mathcal{H}$. We show how Stone's formula for computing the spectral resolution of a Hilbert space self-adjoint operator, can serve as an alternative to the traditional reliance on splitting (or disentanglement) formulas for the operator exponential.

The Weibull distribution is used to describe various observed failures of phenomena and widely used in survival analysis and reliability theory. Sometimes it is very difficult to compute moments of such distributions due to various reasons for e.g. analytical issues, multi parameter cases etc. This study presents the computation of the moments and the expected value of the product of order statistics in the sample from the one-parameter Weibull distribution. An alternative approach in connection to survival function is used to obtain these moments and expected values. In addition the characteristic function of the above distribution is also obtained in the form of gamma functions. Further an illustration is shown to find the first two moments and expected value of the product of order statistics by using this approach.

© 2019, © 2019 American Statistical Association. Consider the model Y = X + ε with X = τ + Z, where τ is an unknown constant (the boundary of X), Z is a random variable defined on R + , ε is a symmetric error, and ε and Z are independent. Based on an iid sample of Y, we aim at identifying and estimating the boundary τ when the law of ε is unknown (apart from symmetry) and in particular its variance is unknown. We propose an estimation procedure based on a minimal distance approach and by making use of Laguerre polynomials. Asymptotic results as well as finite sample simulations are shown. The paper also proposes an extension to stochastic frontier analysis, where the model is conditional to observed variables. The model becomes Y = τ(w1,w2) + Z + ε, where Y is a cost, w 1 are the observed outputs and w 2 represents the observed values of other conditioning variables, so Z is the cost inefficiency. Some simulations illustrate again how the approach works in finite samples, and the proposed procedure is illustrated with data coming from post offices in France. ispartof: Journal Of The American Statistical Association vol:115 issue:529 pages:425-441 status: published

In this paper we consider a telegraph equation with time-dependent coefficients, governing the persistent random walk of a particle moving on the line with a time-varying velocity $c(t)$ and changing direction at instants distributed according to a non-stationary Poisson distribution with rate $\lambda(t)$. We show that, under suitable assumptions, we are able to find the exact form of the probability distribution. We also consider the space-fractional counterpart of this model, finding the characteristic function of the related process. A conclusive discussion is devoted to the potential applications to run-and-tumble models., Comment: Published at https://doi.org/10.15559/18-VMSTA127 in the Modern Stochastics: Theory and Applications (https://vmsta.org/) by VTeX (http://www.vtex.lt/)