Extreme Order Statistics Research Articles

An F1 Beta Distribution with Bathtub Failure Rate Function

SYNOPTIC ABSTRACTBeta distributions are popular models for economic data. In this paper, a new multi modal beta distribution with bathtub shaped failure rate function is introduced. The new distribution is referred to as the F1–beta distribution because its form involves the F1 Appell function of the first kind. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Finally, an application to consumer price indices is illustrated to show that the proposed distribution is a better model to economic data than one based on the standard beta distribution.

On the restricted convergence of generalized extreme order statistics

Generalized order statistics (gos) introduced by Kamps [8] as a unified approach to several models of order random variables (rv’s), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos, included oos and sos, the possible limit distribution functions (df’s) of the maximum gos are obtained in Nasri-Roudsari [10]. In this paper, for this subclass, as the df of the suitably normalized extreme gos converges on an interval [c, d] to one of possible limit df’s of the extreme gos, the continuation of this (weak) convergence on the whole real line to this limit df is proved.

Estimation of the extreme-value index and generalized quantile plots

In extreme-value analysis, a central topic is the adaptive estimation of the extreme-value index γ. Hitherto, most of the attention in this area has been devoted to the case γ>0, that is, when F ¯ is a regularly varying function with index -1/γ. In addition to the well-known Hill estimator, many other estimators are currently available. Among the most important are the kernel-type estimators and the weighted least-squares slope estimators based on the Pareto quantile plot or the Zipf plot, as reviewed by Csorgo and Viharos. Using an exponential regression model (ERM) for spacings between successive extreme order statistics, both Beirlant et al. and Feuerverger and Hall introduced bias-reduced estimators.

Distribution Free Confidence Intervals for Quantiles Based on Extreme Order Statistics in a Multi-Sampling Plan

Distribution Free Confidence Intervals for Quantiles Based on Extreme Order Statistics in a Multi-Sampling Plan

A generalized normal distribution

Undoubtedly, the normal distribution is the most popular distribution in statistics. In this paper, we introduce a natural generalization of the normal distribution and provide a comprehensive treatment of its mathematical properties. We derive expressions for the nth moment, the nth central moment, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, and the asymptotic distribution of the extreme order statistics. We also discuss estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix.

The beta exponential distribution

The exponential distribution is perhaps the most widely applied statistical distribution for problems in reliability. In this note, we introduce a generalization—referred to as the beta exponential distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the beta exponential distribution. We derive expressions for the moment generating function, characteristic function, the first four moments, variance, skewness, kurtosis, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy, the distribution of sums and ratios, and the asymptotic distribution of the extreme order statistics. We also discuss simulation issues, estimation by the methods of moments and maximum likelihood and provide an expression for the Fisher information matrix. We hope that this generalization will attract wider applicability in reliability.

Limit distributions of extreme order statistics of weakly dependent random variables

Limit distributions of extreme order statistics of weakly dependent random variables

Distribution of extreme order statistics in the scheme of allocating particles by groups of random sizes

Distribution of extreme order statistics in the scheme of allocating particles by groups of random sizes

Inequalities for expected extreme order statistics

We obtain one-sided bounds for differences of expected extreme order statistics, when the involved random vectors have (integrable) independent components, and the marginal distributions satisfy a suitable condition. The general results are applied to random samples whose parent distributions are binomial, Poisson, and negative binomial distributions having the same mean. We also derive a method to give lower (upper) bounds for expected maxima (minima) of integrable independent random variables.

A note on the convergence of trivariate extreme order statistics and extension

Necessary and sufficient conditions, under which there exists (at least) a sequence of vectors of real numbers for which the distribution function (d.f.) of any vector of extreme order statistics converges to a nondegenerate limit, are derived. The interesting thing is that these conditions solely depend on the univariate marginals. Moreover, the limit splits into the product of the limit univariate marginals if all the bivariate marginals of the trivariate d.f., from which the sample is drawn, is of negative quadrant dependent random variables (r.v.'s). Finally, all these results are stated for the multivariate extremes with arbitrary dimensions.

Statistical methods in nonlinear dynamics

Sensitivity to initial conditions in nonlinear dynamical systems leads to exponential divergence of trajectories that are initially arbitrarily close, and hence to unpredictability. Statistical methods have been found to be helpful in extracting useful information about such systems. In this paper, we review briefly some statistical methods employed in the study of deterministic and stochastic dynamical systems. These include power spectral analysis and aliasing, extreme value statistics and order statistics, recurrence time statistics, the characterization of intermittency in the Sinai disorder problem, random walk analysis of diffusion in the chaotic pendulum, and long-range correlations in stochastic sequences of symbols.

The exponentiated Gumbel distribution with climate application

The Gumbel distribution is perhaps the most widely applied statistical distribution for climate modeling. In this article we introduce a distribution that generalizes the standard Gumbel distribution in the same way the exponentiated exponential distribution generalizes the standard exponential distribution. We refer to this new distribution as the exponentiated Gumbel distribution. We provide a comprehensive treatment of the mathematical properties of this new distribution and illustrate its use for modeling rainfall data from Orlando, Florida. Among the mathematical properties, we derive the analytical shapes of the corresponding probability density function and the hazard rate function, calculate expressions for the nth moment and the asymptotic distribution of the extreme order statistics, and investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. Copyright © 2005 John Wiley & Sons, Ltd.

On the skew Laplace distribution

A random variable X is said to have the skew-Laplace distribution if its pdf is f(x) = 2g(x)G(λx), where g(·) and G(·), respectively, denote the pdf and the cdf of the Laplace distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et. al [Random Operators and Stochastic Equations, Vol. 10 (2002), pp. 133–140], where expressions for the expectation, variance, skewness and the kurtosis of X are given. But these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X. The properties derived include the k th moment, variance, skewness, kurtosis, moment generating function, characteristic function, cumulant generating function, the k th cumulant, hazard rate function, mean deviation about the mean, mean deviation about the median, Renyi entropy, Shannon's entropy, cumulative residual entropy and the asymptotic distribution of the extreme order statistics. We also consider estimation and simulation issues.

Distribution of extreme order statistics in the scheme of allocating particles by groups of random sizes

We prove a limit theorem which describes the behaviour of extreme order statistics of contents of N cells after allocation of groups of particles of random sizes under some constraints imposed on the distribution of the group size.

Approximating M/G/1 Waiting Time Tail Probabilities

We propose a new approximation formula for the waiting time tail probability of the M/G/1 queue with FIFO discipline and unlimited waiting space. The aim is to address the difficulty of obtaining good estimates when the tail probability has non-exponential asymptotics. We show that the waiting time tail probability can be expressed in terms of the waiting time tail probability of a notional M/G/1 queue with truncated service time distribution plus the tail probability of an extreme order statistic. The Cramér–Lundberg approximation is applied to approximate the tail probability of the notional queue. In essence, our technique extends the applicability of the Cramér–Lundberg approximation to cases where the standard Lundberg condition does not hold. We propose a simple moment-based technique for estimating the parameters of the approximation; numerical results demonstrate that our approximation can yield very good estimates over the whole range of the argument.

On the skew uniform distribution

A random variable X is said to have the skew-uniform distribution if its pdf is ƒ( x ) = 2 g ( x ) G ( λx ), where g (·) and G (·), respectively, denote the pdf and the cdf of the Uniform (− θ , θ ) distribution. This distribution – in spite of its simplicity – appears not to have been studied in detail. The only work that appears to give some details of this distribution is Gupta et al. [ Random Operators and Stochastic Equations , 10 , 2002, 133–140], where expressions for the pdf, moment generating function, expectation, variance, skewness and the kurtosis of X are given. Unfortunately, all of these expressions appear to contain some errors. In this paper, we provide a comprehensive description of the mathematical properties of X . The properties derived include the k th moment, the k th central moment, variance, skewness, kurtosis, moment generating function, characteristic function, hazard rate function, mean deviation about the mean, mean deviation about the median, Rényi entropy, Shannon entropy and the asymptotic distribution of the extreme order statistics. We also consider estimation and simulation issues.

On Rates of Uniform Convergence of Lower Extreme Generalized Order Statistics

Under a von Mises-type condition the joint distribution of suitable normalized lower extreme generalized order statistics converges w.r.t. the variational distance to the asymptotic joint distribution of lower extreme order statistics. Rates of uniform convergence are established. It turns out that the rates of uniform convergence known for ordinary extremes carry over to lower generalized extremes. Finally, models of Weibull type are concerned, where uniform rates are used in connection with model approximations in order to simplify statistical inference.

Estimating catastrophic quantile levels for heavy-tailed distributions

Estimation of the occurrence of extreme events is of prime interest for actuaries. Heavy-tailed distributions are used to model large claims and losses. Within this setting we present a new extreme quantile estimator based on an exponential regression model that was introduced by Feuerverger and Hall [Ann. Stat. 27 (1999) 760] and Beirlant et al. [Extremes 2 (1999) 177]. We also discuss how this approach is to be adjusted in the presence of right censoring. This adaptation can also be linked to robust quantile estimation as this solution is based on a Winsorized mean of extreme order statistics which replaces the classical Hill estimator. We also propose adaptive threshold selection procedures for Weissman’s [J. Am. Stat. Assoc. 73 (1978) 812] quantile estimator which can be used both with and without censoring. Finally some asymptotic results are presented, while small sample properties are compared in a simulation study.

The beta Gumbel distribution

The Gumbel distribution is perhaps the most widely applied statistical distribution for problems in engineering. In this paper, we introduce a generalization—referred to as the beta Gumbel distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of this new distribution. We derive the analytical shapes of the corresponding probability density function and the hazard rate function and provide graphical illustrations. We calculate expressions for the nth moment and the asymptotic distribution of the extreme order statistics. We investigate the variation of the skewness and kurtosis measures. We also discuss estimation by the method of maximum likelihood. We hope that this generalization will attract wider applicability in engineering.

Some further inequalities for univariate moments and some new ones for the covariance

In this paper, some further inequalities for univariate moments are given with particular reference to the expectations of the extreme-order statistics. In addition, some inequalities are obtained for the covariance of two continuous random variables.