Asymptotic distributions for the intervals estimators of the extremal index and the cluster-size probabilities

Under appropriate long range dependence conditions, the point process of exceedances of a stationary sequence weakly converges to a homogeneous compound Poisson point process. This limiting point process can be characterized by the extremal index and the cluster-size probabilities. In this paper we address the problem of estimating these quantities and we consider the intervals estimators introduced in Ferro and Segers [2003. Inference for clusters of extreme values. J. Roy. Statist. Soc. Ser. B 545–556] and in Ferro [2004. Statistical methods for clusters of extreme values. Ph.D. Thesis, Lancaster University]. We establish asymptotic weak convergence to Gaussian random variables and we give their asymptotic variance.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Asymptotic Behavior of Random Time Ruin Probability Under Heavy-Tailed Claim Sizes and Dependence Structure

This article deals with the renewal risk model, in which there exists some asymptotic dependence relation between claim sizes and the inter-arrival times, and claim sizes are subexponential. Under this setting, we investigate the tail behaviour of random time ruin probability as the initial risk reserve x tends to infinity. We obtain the similar asymptotic formula as the previous results.     

Limit Distributions of Random Sums of Z+-Valued Random Variables

In this article, we formulate a transfer theorem in terms of probability generating functions and discuss two approaches to limit distributions of random sums of Z ₊-valued random variables. We then develop Z ₊-valued N-ID and ?-ID laws.     

The Asymptotic Behavior of a Brownian Motion with a Drift from a Random Domain

Consider a Brownian motion with drift starting at an interior point of a random domain D in R ^d+1, d ≥ 1, let τ _D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of log P(τ _D  > T) are given for T → ∞, depending on the shape of the domain D and the order of the drift. The problem is motivated by the model in insurance and early works of Lifshits and Shi. The methods of proof are based on the calculus of variations and early works of Li, Lifshits and Shi in the drift free case.     

Asymptotic approach for a renewal-reward process with a general interference of chance

ABSTRACT

In this study, a renewal-reward process with a discrete interference of chance is constructed and considered. Under weak conditions, the ergodicity of the process X(t) is proved and exact formulas for the ergodic distribution and its moments are found. Within some assumptions for the discrete interference of chance in general form, two-term asymptotic expansions for all moments of the ergodic distribution are obtained. Additionally, kurtosis coefficient, skewness coefficient, and coefficient of variation of the ergodic distribution are computed. As a special case, a semi-Markovian inventory model of type (s, S) is investigated.     

Testing monotonic equilibrium residual entropy of N-State Random Evolution

In this study, we propose an information measure of uncertainty associated with the random equilibrium residual lifetime of a system driven by N-State Random Evolution. A U-statistic test driven by a moment inequality is proposed for testing the hypothesis that the uncertainty of equilibrium remaining life of a system remains unchanged (when system is in the steady state) against the alternative situation when system’s equilibrium residual life has increasing uncertainty over time (i.e., the life distribution has Increasing Equilibrium Residual Entropy property). Some numerical results such as tabulated critical values and empirical power of the proposed test statistic are presented as well.     

Asymptotic variances of maximum likelihood estimator for the correlation coefficient from a BVN distribution with one variable subject to censoring

This paper deals with the problem of estimating the Pearson correlation coefficient when one variable is subject to left or right censoring. In parallel to the classical results on the Pearson correlation coefficient, we derive a workable formula, through tedious computation and intensive simplification, of the asymptotic variances of the maximum likelihood estimators in two cases: (1) known means and variances and (2) unknown means and variances. We illustrate the usefulness of the asymptotic results in experimental designs.     

Parameter Estimation for the Tail Distribution of a Random Sequence

The tail Y_t = X_t – u of a random sequence {X_t ∈ , t ∈ } with identically distributed X_t is approximated by the generalized Pareto distribution according to the extreme value theory, wherein Y_t occurs in clusters because of the dependence in the random sequence. Nevertheless, the parameters of the generalized Pareto distribution are estimated by the same methods as in the case of independent and identically distributed Y_t, provided that there is independence between the clusters of Y_t. The estimation variances and confidence intervals can be estimated by the jackknife method. The approaches are theoretically discussed and verified by extensive numerical researches.     

Asymptotic properties of the maximum-likelihood estimator for a class of birth-and-death processes admitting a unique stationary distribution

The asymptotic properties of the maximum-likelihood estimator of the parameter vector for a class of birth-and-death processes admitting a unique stationary distribution are studied. Also, it is shown that identifiability of the parameter vector with respect to the likelihood implies that the Fisher information matrix is of full rank. Two special cases of biological interest are presented. One of these, the exponential birth-and-death process, is proposed as a more appropriate model of density dependence than the logistic process.     

Large Deviations for Empirical Estimators of the Stationary Distribution of a Semi-Markov Process with Finite State Space

We prove the large deviation principle for empirical estimators of stationary distributions of semi-Markov processes with finite state space, irreducible embedded Markov chain, and finite mean sojourn time in each state. We consider on/off Gamma sojourn processes as an illustrative example, and, in particular, continuous time Markov chains with two states. In the second case, we compare the rate function in this article with the known rate function concerning another family of empirical estimators of the stationary distribution.     

Application of Concomitants of Order Statistics of Independent Non Identically Distributed Random Variables in Estimation

In this article, we obtain expressions for the pdf of a single concomitant of order statistic and the joint pdf of a pair of concomitants of order statistics of independent non identically distributed random variables. Using these expressions, we find the means, variances and covariances of order statistics arising from independent non identically distributed bivariate Pareto distributions. A method of estimation of a common parameter involved in several bivariate Pareto distributions using concomitants of order statistics is also discussed.     

On Robust Analysis of a Normal Location Parameter

Pericchi and Smith considered a normal location parameter problem with double-exponential and Student t prior distributions. These two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem. In this paper we extend the analysis to other scale mixtures of normal distributions, such as the exponential power and the symmetric stable distributions.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

On Estimating the Reduced Second Moment Measure of a Stationary Spatial Point Process

A method is proposed for estimating the covariance structure of a nonparametric estimator for the reduced second moment measure, K(s) , of a homogeneous planar Poisson process. The method relies on the invariance of the reduced second moment measure to random thinning, and the known covariance structure of the estimator under random sampling from a fixed set of points. The possible extension of the method to stationary Cox processes is discussed.     

On a Generalization of the Classical Random Sampling Scheme and Unbiased Estimation

A generalization of the classical random sampling scheme is suggested. Based on the proposed generalization one can derive many new minimum variance unbiased estimators for probabilities, as well as for other functions of unknown parameters, for the multivariate Pólya, the multivariate negative Pólya, the multinomial, the multivariate hypergeometric, the multivariate Poisson, and the Wishart probability distributions.     

On the Moments of a Semi-Markovian Random Walk with Gaussian Distribution of Summands

In this article, a semi-Markovian random walk with delay and a discrete interference of chance (X(t)) is considered. It is assumed that the random variables ζ _n , n = 1, 2,…, which describe the discrete interference of chance form an ergodic Markov chain with ergodic distribution which is a gamma distribution with parameters (α, λ). Under this assumption, the asymptotic expansions for the first four moments of the ergodic distribution of the process X(t) are derived, as λ → 0. Moreover, by using the Riemann zeta-function, the coefficients of these asymptotic expansions are expressed by means of numerical characteristics of the summands, when the process considered is a semi-Markovian Gaussian random walk with small drift β.