Search

We give expansions for the distribution function, density function and moments of the sample minimum when sampling from a distribution on 0 , x 2 that is nearly analytic at zero. Similar results are given for the sample maximum when sampling from a distribution on x 1 , x 2 that is nearly analytic at x 2. When these distributions are analytic, the expansions are in inverse powers of the sample size n. If not, they require a double expansion. [ABSTRACT FROM AUTHOR]

In this paper, we establish the second-order distributional expansions of normalized maxima of n independent observations, where the i th observation follows from a normal copula with its correlation coefficient being a monotone continuous function. These expansions can be used to deduce the convergence rates of distributions of normalized maxima to their limits. [ABSTRACT FROM AUTHOR]

In this paper, we study second order expansions of distributions of maxima of bivariate Gaussian triangular arrays under power normalization. Numerical analysis is given to compare the asymptotic behaviors under power normalization with the asymptotic behaviors under linear normalization derived by Hashorva et al. (2016). [ABSTRACT FROM AUTHOR]

We obtain the analogue of the classical result by Erdös and Kac on the limiting distribution of the maximum of partial sums for exchangeable random variables with zero mean and variance one. We show that, if the conditions of the central limit theorem of Blum et al. hold, the limit coincides with the classical one. Under more general assumptions, the probability of the random variables having conditional negative drift appears in the limit. [ABSTRACT FROM AUTHOR]

We show that the expected value of the descent after the first maximum in a sample of i.i.d. discrete random variables, as the sample size grows, behaves asymptotically up to vanishing terms as the expectation of the maximal value minus the expectation of a sampled random variable, provided the latter is finite. We also show that the expected value after the last maximum exhibits the same behaviour, although it is in general slightly bigger in mean. [ABSTRACT FROM AUTHOR]

Consider a sequence of n observations from an autoregressive process of order 1 with maximum M n and minimum m n . We give their joint cumulative distribution function first in terms of n repeated integrals and then, for the case, where the marginal distribution of the observations is absolutely continuous, as a weighted sum of n th powers of eigenvalues of a certain Fredholm kernel. This enables good approximations for the joint distribution when n repeated integrals is not a practical solution. [ABSTRACT FROM AUTHOR]

We consider samples of geometric random variables and find the average size of the descent after the first and last maximal values. These are asymptotically but not exactly equal, with the descent after the last maximum being slightly larger than that after the first. Thereafter we calculate the probability that the descent after the last maximum is equal to, greater than, or less than the descent after the first maximum. Finally we compute asymptotic expansions for these probabilities. [ABSTRACT FROM AUTHOR]

Motivated by the problem of computing the distribution of the largest distance dmax between n random points on a circle we derive an explicit formula for the moments of the maximal component of a random vector following a Dirichlet distribution with concentration parameters (1,...,1). We use this result to give a new proof of the fact that the law of ndmax - log n converges to a Gumbel distribution as n tends to infinity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we derive the exact uniform convergence rate of the asymmetric normal distribution of the maximum and minimum to its extreme value limit. [Copyright &y& Elsevier]

Abstract: For a skew-normal random sequence, convergence rates of the distribution of its partial maximum to the Gumbel extreme value distribution are derived. The asymptotic expansion of the distribution of the normalized maximum is given under an optimal choice of norming constants. We find that the optimal convergence rate of the normalized maximum to the Gumbel extreme value distribution is proportional to . [Copyright &y& Elsevier]

Abstract: Suppose we observe for in , where are independent and identical random errors with common distribution function . Let . When the upper tail of is of power-type, local power type, gamma type and normal type, we give conditions on the growth of the location and scale trends such that for certain constants and , converges to one of the three standard extreme value distributions. In each case is proportional to the -norm of and does not depend on . Most importantly, trend in scale is shown to dominate trend in location. [Copyright &y& Elsevier]

Abstract: Let be a continuous mean square differentiable stationary Gaussian process. Under some mild restrictions on its correlation function , we prove an almost sure limit theorem for the maximum of the Gaussian process . [Copyright &y& Elsevier]

Abstract: In this short note, we focus on the limiting behaviors of moments for normalized partial maxima of skew-normal samples. Under optimal norming constants, asymptotic expansions for moments of maxima of skew-normal samples are derived. These expansions are used to deduce convergence rates of moments of the normalized maxima to the moments of the corresponding extreme value distribution. [Copyright &y& Elsevier]

Abstract: By using the theory of p-max stable laws, we study the rates of convergence of extremes for general error distribution under power normalization. We derived the exact uniform convergence rate of the distribution of maximum to its extreme value limit. [Copyright &y& Elsevier]

Abstract: A study of the expected value of the maximum of independent, identically distributed (IID) geometric random variables is presented based on the Fourier analysis of the distribution of the fractional part of the maximum of corresponding IID exponential random variables. [Copyright &y& Elsevier]

Abstract: We derive the exact probability density function of the maximum of arbitrary absolutely continuous dependent random variables and of absolutely continuous exchangeable random variables. We show this density is related to the family of fundamental skew distributions. In particular, we examine the case where the random variables have an elliptically contoured distribution. We study some particular examples based on the multivariate normal and multivariate Student t distributions, and discuss numerical computation issues. We illustrate our results on a genetic selection problem and on an autoregressive time series model of order one. [Copyright &y& Elsevier]

Abstract: The maximum of a hazard function is a parameter of great importance in seismicity studies, because it constitutes the maximum risk of occurrence of an earthquake in a given interval of time. By means of kernel nonparametric estimates of the first derivative of the hazard function, we establish uniform convergence properties and asymptotic normality of an estimate of the maximum, in a context of strong mixing dependence. A small simulation study and a practical example show the performance of the proposed estimator in finite samples. [Copyright &y& Elsevier]

Abstract: In a stationary sequence of random variables, high-threshold exceedances may cluster together. Two approximations of such a cluster''s distribution are established. These justify and generalize sampling schemes for clusters of extremes already known for Markov chains. [Copyright &y& Elsevier]

Using Rice''s method to obtain the distribution of the maximum of Gaussian random fields we give the distribution of the length of the projection of a Gaussian variable onto any convex or non-convex spherical cone. [Copyright &y& Elsevier]