Inequalities involving expectations of selected functions in reliability theory to characterize distributions

Recently, authors have studied inequalities involving expectations of selected functions, viz. failure rate, mean residual life, aging intensity function, and log-odds rate which are defined for left truncated random variables in reliability theory to characterize some well-known distributions. However, there has been growing interest in the study of these functions in reversed time (X ? x, instead of X > x) and their applications. In the present work we consider reversed hazard rate, expected inactivity time, and reversed aging intensity function to deal with right truncated random variables and characterize a few statistical distributions.     

A Class of Multivariate Bilateral Selection t Distributions and Its Properties

This article proposes a class of multivariate bilateral selection t distributions useful for analyzing non-normal (skewed and/or bimodal) multivariate data. The class is associated with a bilateral selection mechanism, and it is obtained from a marginal distribution of the centrally truncated multivariate t. It is flexible enough to include the multivariate t and multivariate skew-t distributions and mathematically tractable enough to account for central truncation of a hidden t variable. The class, closed under linear transformation, marginal, and conditional operations, is studied from several aspects such as shape of the probability density function, conditioning of a distribution, scale mixtures of multivariate normal, and a probabilistic representation. The relationships among these aspects are given, and various properties of the class are also discussed. Necessary theories and two applications are provided.     

Some multivariate singular unified skew-normal distributions and their application

Abstract

We introduce here the truncated version of the unified skew-normal (SUN) distributions. By considering a special truncations for both univariate and multivariate cases, we derive the joint distribution of consecutive order statistics X_{(r, ..., r + k)} = (X_(r), ..., X_{(r + K)})^T from an exchangeable n-dimensional normal random vector X. Further we show that the conditional distributions of X_{(r + j, ..., r + k)} given X_{(r, ..., r + j ? 1)}, X_{(r, ..., r + k)} given (X_(r) > t)?and X_{(r, ..., r + k)} given (X_{(r + k)} < t) are special types of singular SUN distributions. We use these results to determine some measures in the reliability theory such as the mean past life (MPL) function and mean residual life (MRL) function.     

Some results on the truncated multivariate t distribution

The use of truncated distributions arises often in a wide variety of scientific problems. In the literature, there are a lot of sampling schemes and proposals developed for various specific truncated distributions. So far, however, the study of the truncated multivariate t (TMVT) distribution is rarely discussed. In this paper, we first present general formulae for computing the first two moments of the TMVT distribution under the double truncation. We formulate the results as analytic matrix expressions, which can be directly computed in existing software. Results for the left and right truncation can be viewed as special cases. We then apply the slice sampling algorithm to generate random variates from the TMVT distribution by introducing auxiliary variables. This strategic approach can result in a series of full conditional densities that are of uniform distributions. Finally, several examples and practical applications are given to illustrate the effectiveness and importance of the proposed results.     

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.     

The role of weighted distributions in stochastic modeling

C. R. Rao pointed out that “The role of statistical methodology is to extract the relevant information from a given sample to answer specific questions about the parent population” and raised the question “What population does a sample represent”? Wrong specification can lead to invalid inference giving rise to a third kind of error. Rao introduced the concept of weighted distributions as a method of adjustment applicable to many situations.

In this paper, we study the relationship between the weighted distributions and the parent distributions in the context of reliability and life testing. These relationships depend on the nature of the weight function and give rise to interesting connections between the different ageing criteria of the two distributions. As special cases, the length biased distribution, the equilibrium distribution of the backward and forward recurrence times and the residual life distribution, which frequently arise in practice, are studied and their relationships with the original distribution are examined. Their survival functions, failure rates and mean residual life functions are compared and some characterization results are established.     

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.     

Inverse moments of discrete distributions

In this paper an expression for the inverse moment of order r is given for the truncated binomial and Poisson distributions. This enables one to obtain inverse moments in a finite series. Some applications and multivariate generalizations are also given. The method also enables one to obtain relations between inverse moments and factorial moments and distributions of sums of variables.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.     

The beta exponentiated Weibull distribution

The Weibull distribution is one of the most important distributions in reliability. For the first time, we introduce the beta exponentiated Weibull distribution which extends recent models by Lee et al. [Beta-Weibull distribution: some properties and applications to censored data, J. Mod. Appl. Statist. Meth. 6 (2007), pp. 173–186] and Barreto-Souza et al. [The beta generalized exponential distribution, J. Statist. Comput. Simul. 80 (2010), pp. 159–172]. The new distribution is an important competitive model to the Weibull, exponentiated exponential, exponentiated Weibull, beta exponential and beta Weibull distributions since it contains all these models as special cases. We demonstrate that the density of the new distribution can be expressed as a linear combination of Weibull densities. We provide the moments and two closed-form expressions for the moment-generating function. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and entropies. The density of the order statistics can also be expressed as a linear combination of Weibull densities. We obtain the moments of the order statistics. The expected information matrix is derived. We define a log-beta exponentiated Weibull regression model to analyse censored data. The estimation of the parameters is approached by the method of maximum likelihood. The usefulness of the new distribution to analyse positive data is illustrated in two real data sets.     

Multivariate Birnbaum–Saunders distribution: properties and associated inference

The univariate fatigue life distribution proposed by Birnbaum and Saunders [A new family of life distributions. J Appl Probab. 1969;6:319–327] has been used quite effectively to model times to failure for materials subject to fatigue and for modelling lifetime data and reliability problems. In this article, we introduce a Birnbaum–Saunders (BS) distribution in the multivariate setting. The new multivariate model arises in the context of conditionally specified distributions. The proposed multivariate model is an absolutely continuous distribution whose marginals are univariate BS distributions. General properties of the multivariate BS distribution are derived and the estimation of the unknown parameters by maximum likelihood is discussed. Further, the Fisher's information matrix is determined. Applications to real data of the proposed multivariate distribution are provided for illustrative purposes.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

A new test for decreasing mean residual lifetimes

The mean residual life of a non negative random variable X with a finite mean is defined by M(t) = E[X ? t|X > t] for t ? 0. One model of aging is the decreasing mean residual life (DMRL): M is decreasing (non increasing) in time. It vastly generalizes the more stringent model of increasing failure rate (IFR). The exponential distribution lies at the boundary of both of these classes. There is a large literature on testing exponentiality against DMRL alternatives which are all of the integral type. Because most parametric families of DMRL distributions are IFR, their relative merits have been compared only at some IFR alternatives. We introduce a new Kolmogorov–Smirnov type sup-test and derive its asymptotic properties. We compare the powers of this test with some integral tests by simulations using a class of DMRL, but not IFR alternatives, as well as some popular IFR alternatives. The results show that the sup-test is much more powerful than the integral tests in all cases.     

Mean residual weighted versus the length-biased Rayleigh distribution

In some statistical applications, data may not be considered as a random sample of the whole population and some subjects have less probability of belonging to the sample. Consequently, statistical inferences for such data sets, usually yields biased estimation. In such situations, the length-biased version of the original random variable as a special weighted distribution often produces better inferences. An alternative weighted distribution based on the mean residual life is suggested to treat the biasedness. The Rayleigh distribution is applied in many real applications, hence the proposed method of weighting is performed to produce a new lifetime distribution based on the Rayleigh model. In addition, statistical properties of the proposed distribution is investigated. A simulation study and a real data set are prepared to illustrate that the mean residual weighted Rayleigh distribution gives a better fit than the original and also the length-biased Rayleigh distribution.     

On bivariate transformation of scale distributions

ABSTRACT

Elsewhere, I have promoted (univariate continuous) “transformation of scale” (ToS) distributions having densities of the form 2g(Π^?1(x)) where g is a symmetric distribution and Π is a transformation function with a special property. Here, I develop bivariate (readily multivariate) ToS distributions. Univariate ToS distributions have a transformation of random variable relationship with Azzalini-type skew-symmetric distributions; the bivariate ToS distribution here arises from marginal variable transformation of a particular form of bivariate skew-symmetric distribution. Examples are given, as are basic properties—unimodality, a covariance property, random variate generation—and connections with a bivariate inverse Gaussian distribution are pointed out.     

On Properties of Reversed Mean Residual Life Order for Weighted Distributions

In this article, first, in order to compare X and X ^w (the weighted version of X with weight function w(·)) according to reversed mean residual life order, we provide an equivalent condition. We then try to provide conditions under which the reversed mean residual life order is preserved by weighted distributions. For this end, we obtain several independent results. Finally, the problem of preservation of increasing reversed mean residual life class under weighting is investigated, as well. Some examples are also given to illustrate the results.     

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.     

Optimal tolerance regions for future regression vector and residual sum of squares of multiple regression model with multivariate spherically contoured errors

This paper considers multiple regression model with multivariate spherically symmetric errors to determine optimal β-expectation tolerance regions for the future regression vector (FRV) and future residual sum of squares (FRSS) by using the prediction distributions of some appropriate functions of future responses. The prediction distribution of the FRV, conditional on the observed responses, is multivariate Student-t distribution. Similarly, the prediction distribution of the FRSS is a beta distribution. The optimal β-expectation tolerance regions for the FRV and FRSS have been obtained based on the F -distribution and beta distribution, respectively. The results in this paper are applicable for multiple regression model with normal and Student-t errors.      

Multivariate distributions involving ratios of normal variables

The papsr considers distributions of collections of ratios of normal variables, The derivation of the joint density is linked to SKI sting literature on absolute, incomplete or truncated moments of multinormals. The distribution function may be expressed as a sum of rectangular multi normal probabilities. When the coefficients of variation of the denominators are close to zero, then a simple transformation of the ratios is approximately inultinormal. An application to Bayesian analysis is included.