Distribution of the product of a pair of independent two-sided power variates

ABSTRACT

The distributions of algebraic functions of random variables are important in theory of probability and statistics and other areas such as engineering, reliability, and actuarial applications, and many results based on various distributions are available in the literature. The two-sided power distribution is defined on a bounded range, and it is a generalization of the uniform, triangular, and power-function probability distributions. This paper gives the exact distribution of the product of two independent two-sided power-distributed random variables in a computable representation. The percentiles of the product are then computed, and a real data application is given.     

A Numerical Study of Small Parameter Behavior of Some Families of Distributions

Limiting distributions play an important role in approximating the exact distributions, especially when they have a rather cumbersome analytic form, or simply when they do not have a closed from. The question that naturally arises is how good the approximation is. In this article, we propose a procedure for the numerical assessment of the “goodness” of some easy-to-calculate limiting distributions, originally proposed in Bar-Lev and Enis, in various cases of the underlying distributions, some of which are inherently computationally challenging. The details of the procedure are provided in three examples. The first example deals with the gamma distributions; the second deals with Bessel distributions related to a symmetric random walk, and the third example deals with positive stable distributions. The details of two additional variations of these examples are also discussed. These examples illustrate the ease with which the limiting approximations could be applied in the various cases, well-demonstrating their computational simplicity and attractiveness.     

Small Sample Tests for Shape Parameters of Gamma Distributions

The introduction of shape parameters into statistical distributions provided flexible models that produced better fit to experimental data. The Weibull and gamma families are prime examples wherein shape parameters produce more reliable statistical models than standard exponential models in lifetime studies. In the presence of many independent gamma populations, one may test equality (or homogeneity) of shape parameters. In this article, we develop two tests for testing shape parameters of gamma distributions using chi-square distributions, stochastic majorization, and Schur convexity. The first one tests hypotheses on the shape parameter of a single gamma distribution. We numerically examine the performance of this test and find that it controls Type I error rate for small samples. To compare shape parameters of a set of independent gamma populations, we develop a test that is unbiased in the sense of Schur convexity. These tests are motivated by the need to have simple, easy to use tests and accurate procedures in case of small samples. We illustrate the new tests using three real datasets taken from engineering and environmental science. In addition, we investigate the Bayes’ factor in this context and conclude that for small samples, the frequentist approach performs better than the Bayesian approach.     

The inverted Nadarajah–Haghighi distribution: estimation methods and applications

The inverted (or inverse) distributions are sometimes very useful to explore additional properties of the phenomenons which non-inverted distributions cannot. We introduce a new inverted model called the inverted Nadarajah–Haghighi distribution which exhibits decreasing and unimodal (right-skewed) density while the hazard rate shapes are decreasing and upside-down bathtub. Our main focus is the estimation (from both frequentist and Bayesian points of view) of the unknown parameters along with some mathematical properties of the new model. The Bayes estimators and the associated credible intervals are obtained using Markov Chain Monte Carlo techniques under squared error loss function. The gamma priors are adopted for both scale and shape parameters. The potentiality of the distribution is analysed by means of two real data sets. In fact, it is found to be superior in its ability to sufficiently model the data as compared to the inverted Weibull, inverted Rayleigh, inverted exponential, inverted gamma, inverted Lindley and inverted power Lindley models.     

The linear combination,product and ratio of Laplace random variables

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this paper, the exact distributions of the linear combination α X+β Y, the product |X Y| and the ratio |X/Y| are derived when X and Y are independent Laplace random variables. The Laplace distribution, being the oldest model for continuous data, has been one of the most popular models for measurement errors in engineering.     

Characterizations and ordering properties based on log-odds functions

In this paper, we obtain some general results on characterizations of probability distributions from relationships between conditional moment, failure rate, and log-odds rate functions. We also study stochastic orders and classes based on the log-odds rate function and some relationships with usual stochastic orderings and classes. Some characterizations and ordering properties are obtained by using weighted distributions.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

Optimal bounds for the mean of the total time on test for distributions with decreasing generalized failure rate

During any life test experiment it is of interest to study potential costs (or profits) of performing the test to the very end. Assuming that these costs are proportional to lifetimes of analysed items the experimenter needs to know the remaining total time on test, having just observed successive failure in the test. We derive sharp upper bounds on the expectation of the remaining total time on test statistic when the underlying distributions have decreasing generalized failure rate with respect to generalized Pareto distributions. In particular we obtain the bounds valid for distributions with decreasing density or failure rate. The results are illustrated with numerical examples.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

The compound class of linear failure rate-power series distributions: Model,properties, and applications

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared using two real datasets.     

Construction of bivariate and multivariate weighted distributions via conditioning

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this article, we propose an alternative method to construct a new family of bivariate and multivariate weighted distributions. For illustrative purposes, some examples of the proposed method are presented. Several structural properties of the bivariate weighted distributions including marginal distributions together with distributions of the minimum and maximum, evaluation of the reliability parameter, and verification of total positivity of order two are also presented. In addition, we provide some multivariate extensions of the proposed models. A real-life data set is used to show the applicability of these bivariate weighted distributions.     

Stochastic models of failure in random environments

Stochastic models of failure modes of frequent occurrence in the engineering sciences are considered. The failure-producing stress environment is modelled as a stationary stochastic process. Using theoretical properties of the sample paths of these processes, failure-time distributions which belong to the Birnbaum-Saunders family are obtained. Several examples of particular engineering relevance are treated.     

Lifetime distributions motivated by series and parallel structures

According to Ross, any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. Motivated by this, we propose new three-parameter lifetime distributions by compounding geometric, power series, and exponential distributions. The distributions can allow for decreasing, increasing, bathtub-shaped, and upside down bathtub-shaped hazard rates. A mathematical treatment of the new distributions is provided including expressions for their density functions, Shannon and Rényi entropies, mean residual life functions, hazard rate functions, quantiles, and moments. The method of maximum likelihood is used for estimating parameters. Five of the new distributions are studied in detail. Finally, two illustrative data examples and a sensitivity analysis are presented.     

Discrete Weibull geometric distribution and its properties

In this article, the discrete analog of Weibull geometric distribution is introduced. Discrete Weibull, discrete Rayleigh, and geometric distributions are submodels of this distribution. Some basic distributional properties, hazard function, random number generation, moments, and order statistics of this new discrete distribution are studied. Estimation of the parameters are done using maximum likelihood method. The applications of the distribution is established using two datasets.     

The Effect of Mis-Specification Between the Lognormal and Weibull Distributions on the Interval Estimation of a Quantile for Complete Data

The lognormal and Weibull distributions are the most popular distributions for modeling lifetime data. In practical applications, they usually fit the data at hand well. However, their predictions may lead to large differences. The main purpose of the present article is to investigate the impacts of mis-specification between the lognormal and Weibull distributions on the interval estimation of a pth quantile of the distributions for complete data. The coverage probabilities of the confidence intervals (CIs) with mis-specification are evaluated. The results indicate that for both the lognormal and the Weibull distributions, the coverage probabilities are significantly influenced by mis-specification, especially for a small or a large p on lower or upper tail of the distributions. In addition, based on the coverage probabilities with correct and mis-specification, a maxmin criterion is proposed to make a choice between these two distributions. The numerical results indicate that for p ≤ 0.05 and 0.6 ≤ p ≤ 0.8, Weibull distribution is suggested to evaluate CIs of a pth quantile of the distributions, while, for 0.2 ≤ p ≤ 0.5 and p = 0.99, lognormal distribution is suggested to evaluate CIs of a pth quantile of the distributions. Besides, for p = 0.9 and 0.95, lognormal distribution is suggested if the sample size is large enough, while, for p = 0.1, Weibull distribution is suggested if the sample size is large enough. Finally, a simulation study is conducted to evaluate the efficiency of the proposed method.     

Generalized Linear Failure Rate Distribution

The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately, the exponential distribution only has constant failure rate and the Rayleigh distribution has increasing failure rate. The linear failure rate distribution generalizes both these distributions which may have non increasing hazard function also. This article introduces a new distribution, which generalizes linear failure rate distribution. This distribution generalizes the well-known (1) exponential distribution, (2) linear failure rate distribution, (3) generalized exponential distribution, and (4) generalized Rayleigh distribution. The properties of this distribution are discussed in this article. The maximum likelihood estimates of the unknown parameters are obtained. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.     

A note on the EM algorithm for estimation in the destructive negative binomial cure rate model

In this note we present a modification in the EM algorithm for the destructive negative binomial cure rate model. This alteration enables us to obtain the estimates of the whole parameter vector from the complete log-likelihood function, avoiding the corresponding observed log-likelihood function, which is more involved. To achieve this goal, we resort to the mixture representation of the negative binomial distribution in terms of the Poisson and gamma distributions.     

Corrected maximum likelihood estimators in heteroscedastic symmetric nonlinear models

In this article, we derive general matrix formulae for second-order biases of maximum likelihood estimators (MLEs) in a class of heteroscedastic symmetric nonlinear regression models, thus generalizing some results in the literature. This class of regression models includes all symmetric continuous distributions, and has a wide range of practical applications in various fields such as engineering, biology, medicine and economics, among others. The variety of distributions with different kurtosis coefficients than the normal may give more flexibility in the choice of an appropriate distribution, particularly to accommodate outlying and influential observations. We derive a joint iterative process for estimating the mean and dispersion parameters. We also present simulation studies for the biases of the MLEs.     

A NEW FAMILY OF NON-NEGATIVE DISTRIBUTIONS

We introduce a new, flexible family of distributions for non‐negative data, defined by means of a quantile function. We describe some properties of this family, and discuss several methods for estimating the parameters. The distribution is applied to an example from environmental engineering.     

Some Relationships Between Skew-Normal Distributions and Order Statistics from Exchangeable Normal Random Vectors

The article explores the relationship between distributions of order statistics from random vectors with exchangeable normal distributions and several skewed generalizations of the normal distribution. In particular, we show that the order statistics of exchangeable normal observations have closed skew-normal distributions, and that the corresponding density function is log-concave when the order statistic is extreme. Special attention is given to the bivariate case, which is related to the univariate skew-normal distribution. The applications discussed focus on the lifetimes of coherent systems.