Extreme value modeling under power normalization

In this paper the mathematical modeling of extremes under power normalization is developed. An estimate of the shape parameter within the generalized extreme value distribution under power normalization is suggested. The statistical inference about the upper tail of a distribution function by using the power normalization is studied. Two models for generalized Pareto distribution under power normalization (GPDP) are given. Estimates for the shape and scale parameters within these GPDP’s are obtained. Finally, a simulation study illustrates and corroborates theoretical results.     

Limit theorems for records from discrete distributions

Weak and strong functional limit theorems are obtained for record values and record epochs in a sequence of independent random variables with common distribution F. The emphasis is on the case in which F is concentrated on the non-negative integers. For contrast, the well-known case of continuous F is also considered. Analogues of results obtained earlier by Resnick, de Haan and the author for continuous F are presented here for F concentrated on the non-negative integers. Also is investigated under which circumstances the latter case is so close to the continuous F case that the resulting limit theorems are the same.     

Some properties of record values coming from the geometric distribution

The independence between spacings of record values and of individual record values is well known when the records come from a geometric distribution. Here we examine the form a function of two record values must have if we require independence from a lower record value. Also similar questions are examined in relation to conditional expectation and conditional variance.     

Limit theorems for numbers of near-records

Observations occurring between successive record times and within a distance a > 0 of the current record value are called near-records. Limit theorems for the number ξ _n (a) of near records are found for cases in which the parent distribution lies in a maximal domain of attraction and a is a function of n. Corollaries are indicated for numbers of near-k-records and sums of near-records. If the parent law is thin-tailed and a is constant, then a centered and normed version of logξ _n (a) has a limit law under appropriate conditions.      

Extreme Values and the Multivariate Compact Law of the Iterated Logarithm

For a sequence of independent identically distributed Euclidean random vectors, we prove a compact Law of the iterated logarithm when finitely many maximal terms are omitted from the partial sum. With probability one, the limiting cluster set of the appropriately operator normed partial sums is the closed unit Euclidean ball. The result is proved under the hypotheses that the random vectors belong to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The integrability condition characterizes how many maximal terms must be omitted from the partial sum sequence.     

Approximations for Bivariate Extreme Values

We study the tail behavior of distributions in the domain of attraction of bivariate extreme value distributions (this includes bivariate extreme value distributions themselves). We provide results on finite approximations of the tail behavior and its analytical shape. The results could form a basis to improve current statistical modeling of bivariate extreme values.     

Probabilistic characterizations of the Generalized Domain of Attraction of the multivariate normal law

For independent identically distributed random vectors,X _i , we give necessary and sufficient probabilistic conditions for their common distribution to belong to the Generalized Domain of Attraction of the multivariate normal law. The first condition says that after projecting onto any direction, , the sum of squares, _i ¹⁼¹ X _i , ², properly normalized, converges to one in probability uniformly over the unit sphere. The second condition says that max X _i , ²/ _n ⁱ⁼¹ X _i , ² converges to zero in probability uniformly over the unit sphere.     

Linear prediction of record values for the two parameter exponential distribution

This paper deals with the problem of predicting thesth record value based on the firstm record values (s>m) when the observations are from the exponential distribution. Various estimates for thesth record value are obtained and their mean square errors are compared.     

Almost sure limit points of and the number of boundary crossings related to SLLN and LIL for record times,inter-record times and the number of record values

In this paper, the almost sure limit points and the moments of the number of boundary crossings of some functions of record times, inter-record times and the frequency of record values of a sequence of independent and identically distributed continuous random variables are studied.     

On the Convergence Rates of Extreme Generalized Order Statistics

A classical result of extreme value theory yields that in case of a linear normalization three possible types of limit distributions are possible. As proved recently a similar classification of the limit distributions holds for extreme generalized order statistics which provide a general concept of ordered random variables. In this paper, we derive results for the convergence rates of the nth and (n-r+1)st generalized order statistic, respectively. It turns out that the rate is highly influenced by the choice of the normalizing sequence. Moreover, we show that a uniform bound of order 1/n holds for underlying generalized Pareto distributions, whereas for the standard normal distribution the convergence might be very slow. Similar results for ordinary order statistics are included.     

On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

The paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987,Stochastic Process. Appl.2583–93], the criteria are easily applicable even when the marginal tails are not Pareto-like.     

一类马尔可夫序列最大值的极限律

主要研究一类马尔可夫序列{Xn,n≥0}的最大值的极限分布.导出了这类序列最大值和最小值的分布表达式,利用经典极值理论,建立了规范化最大值max{X0,X1,…,Xn}与i.i.d序列{ξn,n≥1}的规范化最大值max{1ξ,2ξ,…,ξn+1}具有相同极限律的条件.     

Ratio of Generalized Hill’s estimator and its asymptotic normality theory

We present a statistical process depending on a continuous time parameter τ whose each margin provides a Generalized Hill’s estimator. In this paper, the asymptotic normality of the finite-dimensional distributions of this family are completely characterized for τ > 1/2 when the underlying distribution function lies on the maximum domain of attraction. The ratio of two different margins of the statistical process characterizes entirely the whole domain of attraction. Its asymptotic normality is also studied. The results permit in general to build a new family of estimators for the extreme value index whose asymptotic properties can be easily derived. For example, we give a new estimate of the Weibull extreme value index and we study its consistency and its asymptotic normality.      

Limit theorems for power variations of ambit fields driven by white noise

We study the asymptotics of lattice power variations of two-parameter ambit fields driven by white noise. Our first result is a law of large numbers for power variations. Under a constraint on the memory of the ambit field, normalized power variations converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This result holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asymptotic behavior. Our second result is a stable central limit theorem for thinned power variations.     

Limit theorems for probability measures on simply connected nilpotent lie groups

LetX ₁,X ₂,... be i.i.d. random variables with values in a simply connected nilpotent Lie groupG. Assume the laws of to be weakly convergent to a probability measure onG, _n Aut(G), and (k _n)_n strictly increasing in. In this paper we want to characterize the possible limit laws. We obtain that every limit law is continuously embeddable and we prove a kind of functional limit theorem. Further, we study the connections between two different concepts of stability (resp. semistability) and limit laws. Finally, we describe the various domains of attraction of measures (resp. of convolution semigroups).     

Some Invariance Principles for Random Vectors in the Generalized Domain of Attraction of the Multivariate Normal Law

For independent identically distributed random vectors belonging to the generalized Domain of Attraction of the multivariate normal law, we define two partial sum processes analogous to that of Donsker's Theorem. We prove that each converges in distribution to a Brownian Motion in the space of continuous functions. One process uses nonrandom operator normalization, and the other is a studentization of the first, using normalization by the empirical covariance operator.     

Limit theorems for stationary random evolutions

On a separable Banach space, let A(ξ₁),A(ξ₂),... be a strictly stationary sequence of infinitesimal operators, centered so that EA(ξ_i) = 0, i = 1,2,.... This paper characterizes the limit of the random evolutions

as the solution to a martingale problem. This work is a direct extension of previous work on i.i.d. random evolutions.     

A Compact Law of the Iterated Logarithm for Random Vectors in the Generalized Domain of Attraction of the Multivariate Gaussian Law

For a sequence of independent identically distributed random vectors, we prove that the limiting cluster set of the appropriately operator normed partial sums is, with probability one, the closed unit euclidean ball. The result is proved under the hypotheses that the law of the random vectors belongs to the Generalized Domain of Attraction of the multivariate Gaussian law and satisfy a mild integrability condition. The two conditions together are still weaker than finite second normed moment and are necessary and sufficient.     

Bayesian inference using record values from Rayleigh model with application

In this article, based on a set of upper record values from a Rayleigh distribution, Bayesian and non-Bayesian approaches have been used to obtain the estimators of the parameter, and some lifetime parameters such as the reliability and hazard functions. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX and general entropy (GE)) loss functions. These estimators are derived using the informative and non-informative prior distributions for σ. We compare the performance of the presented Bayes estimators with known, non-Bayesian, estimators such as the maximum likelihood (ML) and the best linear unbiased (BLU) estimators. We show that Bayes estimators under the asymmetric loss functions are superior to both the ML and BLU estimators. The highest posterior density (HPD) intervals for the Rayleigh parameter and its reliability and hazard functions are presented. Also, Bayesian prediction intervals of the future record values are obtained and discussed. Finally, practical examples using real record values are given to illustrate the application of the results.