Weibull tail-distributions revisited: A new look at some tail estimators

In this paper, we propose to include Weibull tail-distributions in a more general family of distributions. In particular, the considered model also encompasses the whole Fréchet maximum domain of attraction as well as log-Weibull tail-distributions. The asymptotic normality of some tail estimators based on the log-spacings between the largest order statistics is established in a unified way within the considered family. This result permits to understand the similarity between most estimators of the Weibull tail-coefficient and the Hill estimator. Some different asymptotic properties, in terms of bias, rate of convergence, are also highlighted.     

Pitfalls in using Weibull tailed distributions

By assuming that the underlying distribution belongs to the domain of attraction of an extreme value distribution, one can extrapolate the data to a far tail region so that a rare event can be predicted. However, when the distribution is in the domain of attraction of a Gumbel distribution, the extrapolation is quite limited generally in comparison with a heavy tailed distribution. In view of this drawback, a Weibull tailed distribution has been studied recently. Some methods for choosing the sample fraction in estimating the Weibull tail coefficient and some bias reduction estimators have been proposed in the literature. In this paper, we show that the theoretical optimal sample fraction does not exist and a bias reduction estimator does not always produce a smaller mean squared error than a biased estimator. These are different from using a heavy tailed distribution. Further we propose a refined class of Weibull tailed distributions which are more useful in estimating high quantiles and extreme tail probabilities.     

A local moment type estimator for an extreme quantile in regression with random covariates

A conditional extreme quantile estimator is proposed in the presence of random covariates. It is based on an adaptation of the moment estimator introduced by Dekkers et al. (1989 Dekkers, A.L.M., Einmahl, J.H.J., de Haan, L. (1989). A moment estimator for the index of an extreme-value distribution. Ann. Statist. 17:1833–1855.[Crossref], [Web of Science ®] , [Google Scholar]) in the classical univariate setting, and thus it is valid in the domain of attraction of the extreme value distribution, i.e., whatever the sign of the extreme value index is. Asymptotic normality of the estimator is established under suitable assumptions, and its finite sample behavior is evaluated with a small simulation study, where a comparison with an alternative estimator already proposed in the literature is provided. An illustration to a real dataset concerning the world catalogue of earthquake magnitudes is also proposed.     

Application of extreme value theory to the bec family of distributions

Arnold and Strauss (1988) derived a family of bivariate life distributions having the property that the conditional distributions are exponential. Asymptotic distributions for the marginal and bivariate extremes for this family of distributions are derived employing the asymptotic theory of extreme order statistics.     

Asymptotic distributions of order statistics and record values arising from the class of beta-generated distributions

In this paper, we show that both the class of beta-generated distributions G_F and its base distribution F belong to the same domain of maximal (or minimal or upper record value or lower record value) attraction. Moreover, it is shown that the weak convergence of any non-extreme order statistic (central or intermediate order statistic), based on a base distribution F, to a non-degenerate limit type implies the weak convergence of G_F to a non-degenerate limit type. The relations between the two limit types are deduced.     

Estimation of quantiles of exponential and double exponential distributions based on two order statistics

Asymptotically best linear unbiased estimators (ABLUE) of quantiles, x^., in the two-parameter (location-scale) exponential and double exponential families are obtained as linear combinations of two suitably chosen order statistics. Exact formulae for the linear combinations are given as functions of £. The derived estimators in both cases compare favorably with the usual nonparametric estimator. Also, in the exponential case the derived estimator compares favorably with the Sarhan-Greenberg BLUE based on a complete sample     

Non‐parametric Estimation of Extreme Risk Measures from Conditional Heavy‐tailed Distributions

In this paper, we introduce a new risk measure, the so‐called conditional tail moment. It is defined as the moment of order a ≥ 0 of the loss distribution above the upper α‐quantile where α ∈ (0,1). Estimating the conditional tail moment permits us to estimate all risk measures based on conditional moments such as conditional tail expectation, conditional value at risk or conditional tail variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where α ↓0 is no longer fixed). It is moreover assumed that the loss distribution is heavy tailed and depends on a covariate. The estimation method thus combines non‐parametric kernel methods with extreme‐value statistics. The asymptotic distribution of the estimators is established, and their finite‐sample behaviour is illustrated both on simulated data and on a real data set of daily rainfalls.     

Re-parameterization of multinomial distributions and diversity indices

It is shown in this paper that the parameters of a multinomial distribution may be re-parameterized as a set of generalized Simpson's diversity indices. There are two important elements in the generalization: (1) Simpson's diversity index is extended to populations with infinite species; (2) weighting schemes are incorporated. A class of unbiased estimators for the generalized Simpson's biodiversity indices is proposed. Asymptotic normality is established for the estimators. Both the unbiasedness and the asymptotic normality of the estimators hold for all three cases of the number of species in the population: infinite, finite and known, and finite but unknown. In the case of a population with a finite number of species, known or unknown, it is also established that the proposed estimators are uniformly minimum variance unbiased and are asymptotically efficient.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

Bootstrap confidence intervals in mixtures of discrete distributions

The problem of building bootstrap confidence intervals for small probabilities with count data is addressed. The law of the independent observations is assumed to be a mixture of a given family of power series distributions. The mixing distribution is estimated by nonparametric maximum likelihood and the corresponding mixture is used for resampling. We build percentile-t and Efron percentile bootstrap confidence intervals for the probabilities and we prove their consistency in probability. The new theoretical results are supported by simulation experiments for Poisson and geometric mixtures. We compare percentile-t and Efron percentile bootstrap intervals with eight other bootstrap or asymptotic theory based intervals. It appears that Efron percentile bootstrap intervals outperform the competitors in terms of coverage probability and length.     

Estimation and Testing in Elliptical Functional Measurement Error Models

The purpose of this article is to investigate estimation and hypothesis testing by maximum likelihood and method of moments in functional models within the class of elliptical symmetric distributions. The main results encompass consistency and asymptotic normality of the method of moments estimators. Also, the asymptotic covariance matrix of the maximum likelihood estimator is derived, extending some existing results in elliptical distributions. A measure of asymptotic relative efficiency is reported. Wald-type statistics are considered and numerical results obtained by Monte Carlo simulation to investigate the performance of estimators and tests are provided for Student-t and contaminated normal distributions. An application to a real dataset is also included.     

Conditional Maximum Likelihood and Interval Estimation for Two Weibull Populations under Joint Type-II Progressive Censoring

Comparative lifetime experiments are important when the object of a study is to determine the relative merits of two competing duration of life products. This study considers the interval estimation for two Weibull populations when joint Type-II progressive censoring is implemented. We obtain the conditional maximum likelihood estimators of the two Weibull parameters under this scheme. Moreover, simultaneous approximate confidence region based on the asymptotic normality of the maximum likelihood estimators are also discussed and compared with two Bootstrap confidence regions. We consider the behavior of probability of failure structure with different schemes. A simulation study is performed and an illustrative example is also given.     

An Elementary Introduction to Maximum Likelihood Estimation for Multinomial Models: Birch's Theorem and the Delta Method

A fairly complete introduction to the large sample theory of parametric multinomial models, suitable for a second-year graduate course in categorical data analysis, can be based on Birch's theorem (1964) and the delta method (Bishop, Fienberg, and Holland 1975). I present an elementary derivation of a version of Birch's theorem using the implicit function theorem from advanced calculus, which allows the presentation to be relatively self-contained. The use of the delta method in deriving asymptotic distributions is illustrated by Rao's (1973) result on the distribution of standardized residuals, which complements the presentation in Bishop, Fienberg, and Holland. The asymptotic theory is illustrated by two examples.     

On the Rate of Convergence of STSD Extremes

Denote M _n the largest of n independent STSD variables. This article shows the rate of convergence of (M _n  ? b _n )/a _n to the extreme value distribution exp (?e ^?x ) which is characterized by the supremum metric.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

Domains of attraction of asymptotic distributions of extreme generalized order statistics

Abstract

In extreme value theory for ordinary order statistics, there are many results that characterize the domains of attraction of the three extreme value distributions. In this article, we consider a subclass of generalized order statistics for which also three types of limit distributions occur. We characterize the domains of attraction of these limit distributions by means of necessary and/or sufficient conditions for an underlying distribution function to belong to the respective domain of attraction. Moreover, we compare the domains of attraction of the limit distributions for extreme generalized order statistics with the domains of attraction of the extreme value distributions.     

Asymptotic Normality of a Kernel Conditional Quantile Estimator Under Strong Mixing Hypothesis and Left-Truncation

We consider the estimation of the conditional quantile when the interest variable is subject to left truncation. Under regularity conditions, it is shown that the kernel estimate of the conditional quantile is asymptotically normally distributed, when the data exhibit some kind of dependence. We use asymptotic normality to construct confidence bands for predictors based on the kernel estimate of the conditional median.     

Probability density function of quotient of order statistics from the pareto,power and weibull distributions

This paper deals with the probability density functions of quotient of order statistics. We use the Mellin transform technique, to find the distribution of the quotient Z= X/Xwhere X.,X(i < j) are the ith and jth order statistics from the Pareto, Power and Weibull distributions     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.