Recurrence relations between product moments of order statistics

A general result for obtaining recurrence relations between product moments of order statistics is established and this result is used to determine the recurrence relations between product moments of some doubly truncated distributions. The examples considered are Weibull, exponential, Pareto, power function and Cauchy distributions.     

Bayesian predictive density of order statistics based on finite mixture models

Bayesian predictive density functions, which are necessary to obtain bounds for predictive intervals of future order statistics, are obtained when the population density is a finite mixture of general components. Such components include, among others, the Weibull (exponential and Rayleigh as special cases), compound Weibull (three-parameter Burr type XII), Pareto, beta, Gompertz and compound Gompertz distributions. The prior belief of the experimenter is measured by a general distribution that was suggested by AL-Hussaini (J. Statist. Plann. Inf. 79 (1999b) 79). Applications to finite mixtures of Weibull and Burr type XII components are illustrated and comparison is made, in the special cases of the exponential and Pareto type II components, with previous results.     

Recurrence relations for moments of progressive type-II right censored order statstics from burr distribution

In this paper some recurrence relations between moments of progressive Type-II right censored order statistics from doubly truncated Burr distribution are established. These recurrence relations would enable one to obtain all the single and product moments of Burr progressive Type-II right censored order statistics in a simple recursive manner.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

Moments of order statistics from parabolic and skewed distributions and a characterization of weibull distribution

In this paper the single and product moments of order statistics from doubly truncated parabolic and skewed distributions have been obtained. Also the Weibull distribution has been characterized through the properties of order statistics.     

The Kumaraswamy Burr XII distribution: theory and practice

For the first time, a five-parameter distribution, called the Kumaraswamy Burr XII (KwBXII) distribution, is defined and studied. The new distribution contains as special models some well-known distributions discussed in lifetime literature, such as the logistic, Weibull and Burr XII distributions, among several others. We obtain the complete moments, incomplete moments, generating and quantile functions, mean deviations, Bonferroni and Lorenz curves and reliability of the KwBXII distribution. We provide two representations for the moments of the order statistics. The method of maximum likelihood and a Bayesian procedure are adopted for estimating the model parameters. For different parameter settings and sample sizes, various simulation studies are performed and compared to the performance of the KwBXII distribution. Three applications to real data sets demonstrate the usefulness of the proposed distribution and that it may attract wider applications in lifetime data analysis.     

Step partially accelerated life tests under finite mixture models

In this paper, step partially accelerated life tests are considered when the lifetime of an item under use condition follows a finite mixture of distributions. The analysis is performed when each of the components follows a general class of distributions, which includes, among others, the Weibull, compound Weibull (or three-parameter Burr type XII), power function, Gompertz and compound Gompertz distributions. Based on type-I censoring, the maximum likelihood estimates (MLEs) of the mixing proportions, scale parameters and acceleration factor are obtained. Special attention is paid to a mixture of two exponential components. Simulation results are obtained to study the precision of MLEs.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Current records and record range with some applications

In a sequence of independent and identically distributed (iid) random variables, the upper (lower) current records and record range are studied. We derive general recurrence relations between the single and product moments for the upper and lower current records based on Weibull and positive Weibull distributions, as well as Pareto and negative Pareto distributions, respectively. Moreover, some asymptotic results for general current records are established. In addition, a recurrence relation and an explicit formula for the moments of record range based on the exponential distribution are given. Finally, numerical examples are presented to illustrate and corroborate theoretical results.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

Moments of truncated normal/independent distributions

In this work we have considered the problem of finding the moments of a doubly truncated member of the class of normal/independent distributions. We obtained a general result and then use it to derive the moments in the case of doubly truncated versions of Pearson type VII distribution, slash distribution, contaminated normal distribution, double exponential distribution and variance gamma distribution. We also give an application of some actuarial data.     

Moments of order statistics from a non-overlapping mixture model with applications to truncated laplace distribution

We employ two different approaches to derive single and product moments of order statistics from a truncated Laplace distribution. A direct evaluation method establishes recurrence relations whereas the more general non-overlapping mixture model incorporates the truncated Laplace distribution as a special case. The results are thereafter applied to estimate location and scale parameters of such distributions.     

Two multivariate generalized beta families

A multivariate generalized beta distribution is introduced that extends the univariate generalized beta distribution and includes many multivariate distributions, such as the multivariate beta of the first and second kind, the generalized gamma, and the Burr and Dirichlet distributions as special and limiting cases. These interrelationships can be illustrated using a distributional family tree. The corresponding marginal distributions are univariate generalized beta distributions and their special cases. Selected expressions for the moments are reported, and an application to the joint distribution of income and wealth is presented. A simple transformation of the multivariate generalized beta distribution leads to what will be referred to as a multivariate exponential generalized beta distribution, which includes a multivariate form of the logistics and Burr distributions as special cases.     

The four-parameter Burr XII distribution: Properties,regression model,and applications

This paper introduces a new four-parameter lifetime model called the Weibull Burr XII distribution. The new model has the advantage of being capable of modeling various shapes of aging and failure criteria. We derive some of its structural properties including ordinary and incomplete moments, quantile and generating functions, probability weighted moments, and order statistics. The new density function can be expressed as a linear mixture of Burr XII densities. We propose a log-linear regression model using a new distribution so-called the log-Weibull Burr XII distribution. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimation are discussed. We prove empirically the importance and flexibility of the new model in modeling various types of data.     

On distributions Of generalized order statistics

In a wide subclass of generalized order statistics, representations of marginal density and distribution functions are developed. The results are applied to obtain several relations, such as recurrence relations, and explicit expressions for the moments of generalized order statistics from Pareto, power function and Weibull distributions Moreover, characterizations of exponential distributions are shown by means of a distributional identity as well as by* an identity of expectations involving a subrange and a corresponding generalized order statistic.     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

Recurrence relations for the moments of truncated multinormal distribution

In this paper some recurrence relations of moments of doubly truncated multivariate normal distribution are obtained. The bivariate case is given as an example and some applications are indicated.     

Bayesian estimation and prediction for some life distributions based on record values

Some statistical data are most easily accessed in terms of record values. Examples include meteorology, hydrology and athletic events. Also, there are a number of industrial situations where experimental outcomes are a sequence of record-breaking observations. In this paper, Bayesian estimation for the two parameters of some life distributions, including Exponential, Weibull, Pareto and Burr type XII, are obtained based on upper record values. Prediction, either point or interval, for future upper record values is also presented from a Bayesian view point. Some of the non-Bayesian results can be achieved as limiting cases from our results. Numerical computations are given to illustrate the results.     

Moment expressions and summary statistics for the complete and truncated weibull distribution

Traditionally, the moments of the Weibull distribution have been calculated using the standard Weibull (Johnson and Kotz, 1970) . This article will expand on that idea and cover the truncated cases for the standard Weibull distributions. Also, the same techniques used for the standard form will be used to derive the moment expressions for the three-parameter complete and truncated Weibull distributions. The summary statistics are then calculated from the moment expressions. Weibull moments involve the gamma and incomplete gamma functions.