The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

Robust nonparametric estimation of the intensity function of point data

Intensity functions—which describe the spatial distribution of the occurrences of point processes—are useful for risk assessment. This paper deals with the robust nonparametric estimation of the intensity function of space–time data from events such as earthquakes. The basic approach consists of smoothing the frequency histograms with the local polynomial regression (LPR) estimator. This method allows for automatic boundary corrections, and its jump-preserving ability can be improved with robustness. We derive a robust local smoother from the weighted-average approach to M-estimation and we select its bandwidths with robust cross-validation (RCV). Further, we develop a robust recursive algorithm for sequential processing of the data binned in time. An extensive application to the Northern California earthquake catalog in the San Francisco, CA, area illustrates the method and proves its validity.     

Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the Associated Pseudo-truncated Weibull

Two important wood properties are stiffness (modulus of elasticity or MOE) and bending strength (modulus of rupture or MOR). In the past, MOE has often been modeled as a Gaussian and MOR as a lognormal or a two or three parameter Weibull. It is well known that MOE and MOR are positively correlated. To model the simultaneous behavior of MOE and MOR for the purposes of wood system reliability calculations, we introduce a bivariate Gaussian–Weibull distribution and the associated pseudo-truncated Weibull. We use asymptotically efficient likelihood methods to obtain an estimator of the parameter vector of the bivariate Gaussian–Weibull, and then obtain the asymptotic distribution of this estimator.     

On convergence of the distributions of random sequences with independent random indexes to variance–mean mixtures

We prove a transfer theorem for random sequences with independent random indexes in the double array limit setting under relaxed conditions. We also prove its partial inverse providing the necessary and sufficient conditions for the convergence of randomly indexed random sequences. Special attention is paid to the case where the elements of the basic double array are formed as cumulative sums of independent not necessarily identically distributed random variables. Using simple moment-type conditions we prove the theorem on convergence of the distributions of such sums to normal variance–mean mixtures.     

An Efficient Method of Parameter and Quantile Estimation for the Three-Parameter Weibull Distribution Based on Statistics Invariant to Unknown Location Parameter

The three-parameter Weibull distribution is widely used in life testing and reliability analysis. In this article, we propose an efficient method for the estimation of parameters and quantiles of the three-parameter Weibull distribution, which avoids the problem of unbounded likelihood, by using statistics invariant to unknown location. Through a Monte Carlo simulation study, we show that the proposed method performs well compared to other prominent methods based on bias and MSE. Finally, we present two illustrative examples.     

Non parametric estimation of the structural expectation of a stochastic increasing function

This article introduces a non parametric warping model for functional data. When the outcome of an experiment is a sample of curves, data can be seen as realizations of a stochastic process, which takes into account the variations between the different observed curves. The aim of this work is to define a mean pattern which represents the main behaviour of the set of all the realizations. So, we define the structural expectation of the underlying stochastic function. Then, we provide empirical estimators of this structural expectation and of each individual warping function. Consistency and asymptotic normality for such estimators are proved.     

Tests of significance for the odds ratio in a 2×2 table based on the ratio of two independent F-variates

Following Gart (1966) a test of significance for the odds ratio in a 2×2 table is developed based on a semi-empirical method of approximating discrete distributions by their continuous analogues. The distribution of the test statistic (W), the ratio of two independent F-variates, is derived. This approximate technique is compared with the "exact" test, uncorrected X test, and a normal approximation based on lnW.     

On the strong convergence for weighted sums of ρ *-mixing random variables

A complete convergence result is obtained for weighted sums of identically distributed ρ ^*-mixing random variables with E|X ₁| ^α log(1 + |X ₁|) < ∞ for some 0 < α ≤ 2. This result partially extends the result of Sung (Stat Papers 52: 447–454, 2011) for negatively associated random variables to ρ ^*-mixing random variables. It also settles the open problem posed by Zhou et al. (J Inequal Appl, 2011, doi:10.1155/2011/157816).     

Bayesian estimation of quantile distributions

Use of Bayesian modelling and analysis has become commonplace in many disciplines (finance, genetics and image analysis, for example). Many complex data sets are collected which do not readily admit standard distributions, and often comprise skew and kurtotic data. Such data is well-modelled by the very flexibly-shaped distributions of the quantile distribution family, whose members are defined by the inverse of their cumulative distribution functions and rarely have analytical likelihood functions defined. Without explicit likelihood functions, Bayesian methodologies such as Gibbs sampling cannot be applied to parameter estimation for this valuable class of distributions without resorting to numerical inversion. Approximate Bayesian computation provides an alternative approach requiring only a sampling scheme for the distribution of interest, enabling easier use of quantile distributions under the Bayesian framework. Parameter estimates for simulated and experimental data are presented.     

GEE estimation of the covariance structure of a bivariate panel data model with an application to wage dynamics and the incidence of profit-sharing in West Germany

We propose a generalized estimating equations (GEE) approach to the estimation of the mean and covariance structure of bivariate time series processes of panel data. The one-step approach allows for mixed continuous and discrete dependent variables. A Monte Carlo Study is presented to compare our particular GEE estimator with more standard GEE-estimators. In the empirical illustration, we apply our estimator to the analysis of individual wage dynamics and the incidence of profit-sharing in West Germany. Our findings show that time-invariant unobserved individual ability jointly influences individual wages and participation in profit sharing schemes.     

Point estimation of the stress–strength reliability parameter for parallel system with independent and non-identical components

In this article, the estimation problem of the multicomponent stress–strength reliability parameter is considered where the stress and the strength systems have arbitrary fixed numbers of independent and non-identical parallel components. It is assumed that the distribution functions of the stress and the strength components satisfy the proportional reversed hazard rate model. The study is done in more details when the baseline distributions are exponential. Maximum likelihood and uniformly minimum variance unbiased estimators are obtained and compared. Also, Bayes and empirical Bayes estimators are discussed and Monte Carlo simulations are carried out to compare their performances.     

The Maximum Probability 2 × c Contingency Tables and the Maximum Probability Points of the Multivariate Hypergeometric Distribution

Abstract

The problem of obtaining the maximum probability 2 × c contingency table with fixed marginal sums, R  = (R ₁, R ₂) and C  = (C ₁, … , C _c ), and row and column independence is equivalent to the problem of obtaining the maximum probability points (mode) of the multivariate hypergeometric distribution MH(R ₁; C ₁, … , C _c ). The most simple and general method for these problems is Joe's (Joe, H. (1988 Joe, H. 1988. Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth., 17(11): 3677–3685. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). Extreme probabilities for contingency tables under row and column independence with application to Fisher's exact test. Commun. Statist. Theory Meth. 17(11):3677–3685.) In this article we study a family of MH's in which a connection relationship is defined between its elements. Based on this family and on a characterization of the mode described in Requena and Martín (Requena, F., Martín, N. (2000 Requena, F. and Martín, N. 2000. Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett., 50: 39–47.  [Google Scholar]). Characterization of maximum probability points in the multivariate hypergeometric distribution. Statist. Probab. Lett. 50:39–47.), we develop a new method for the above problems, which is completely general, non recursive, very simple in practice and more efficient than the Joe's method. Also, under weak conditions (which almost always hold), the proposed method provides a simple explicit solution to these problems. In addition, the well-known expression for the mode of a hypergeometric distribution is just a particular case of the method in this article.     

A note on expressing wilks’ a as the product of independent beta variables

The purpose of this note is to show that the null distribution of Wilks’ A can be expressed as the product of p independent beta variables regardless of the relationship of p and q, where p is the number of variables and q is the degrees of freedom associated with the hypothesis.     

Large deviations for sum of UEND and φ-mixing random variables with heavy tails

Abstract

In this article, we obtain the large deviations for sum of non identically distributed random variables which are upper extended negative dependent and φ-mixing.     

Marginal-sum and maximum-likelihood estimation in a multiplicative tariff

It is well known that, for a multiplicative tariff with independent Poisson distributed claim numbers in the different tariff cells, the maximum-likelihood estimators of the parameters satisfy the marginal-sum equations. In the present paper we show that this is also true under the more general assumption that the claim numbers of the different cells arise from the decomposition of a collective model for the whole portfolio of risks. In this general setting, the claim numbers of the different cells need not be independent and need not be Poisson distributed.     

Precise large deviations for sums of WUOD and φ-mixing random variables with dominated variation

In this paper, we investigate the precise large deviations for sums of not identical distributed random variables which are widely upper orthant dependent (WUOD, in short) and φ-mixing. The asymptotic relations for sums of random variables with dominated variation are obtained.     

Empirical bayes estimation of the log odds ratio in 2×2 contingency tables

We consider a sequence of contingency tables whose cell probabilities may vary randomly. The distribution of cell probabilities is modelled by a Dirichlet distribution. Bayes and empirical Bayes estimates of the log odds ratio are obtained. Emphasis is placed on estimating the risks associated with the Bayes, empirical Bayes and maximum lilkelihood estimates of the log odds ratio.     

On the exact distribution of the sum of the largest n−k out of n normal random variables with differing mean values

Motivated by an application in Electrical Engineering, we derive the exact distribution of the sum of the largest n?k out of n normally distributed random variables, with differing mean values. Comparisons are made with two normal approximations to this distribution—one arising from the asymptotic negligibility of the omitted order statistics and one from the theory of L-statistics. The latter approximation is found to be in excellent agreement with the exact distribution.     

A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.