On bivariate ageing properties of exchangeable Farlie–Gumbel–Morgenstern distributions

This paper deals with bivariate Farlie–Gumbel–Morgenstern distributions. We build the TP2 (RR2) property of the residual lifetime and study the evolution of the dependence of the residual lifetime at large age. Also, we derive the aging property in the sense of the upper orthant order. Furthermore, in the context of IFR (DFR) marginals the residual lifetime at the age above the median is found to be increasing (decreasing) in the upper orthant order.     

Concomitants of generalized order statistics from Farlie–Gumbel–Morgenstern distributions

Generalized order statistics constitute a unified model for ordered random variables that includes order statistics and record values among others. Here, we consider concomitants of generalized order statistics for the Farlie–Gumbel–Morgenstern bivariate distributions and study recurrence relations between their moments. We derive the joint distribution of concomitants of two generalized order statistics and obtain their product moments. Application of these results is seen in establishing some well known results given separately for order statistics and record values and obtaining some new results.     

Estimation and goodness-of-fit procedures for Farlie–Gumbel–Morgenstern bivariate copula of order statistics

In this study, we provide the Farlie–Gumbel–Morgenstern bivariate copula of rth and sth order statistics. The main emphasis in this study is on the inference procedure which is based on the maximum pseudo-likelihood estimate for the copula parameter. As for the methodology, goodness-of-fit test statistic for copulas which is based on a Cramér–von Mises functional of the empirical copula process is applied for selecting an appropriate model by bootstrapping. An application of the methodology to simulated data set is also presented.     

Various measures of dependence of a new asymmetric generalized Farlie–Gumbel–Morgenstern copulas

ABSTRACT

In this paper, we discuss an asymmetric extension of Farlie–Gumbel–Morgenstern copulas studied by several authors and obtain the range of the parameter. We derive an expression for regression function and the properties of these copulas are studied in detail. Also, explicit expressions for various measures of association are obtained. These measures are numerically compared for some particular parametric values of the copulas.     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.     

Information Properties for Concomitants of Order Statistics in Farlie–Gumbel–Morgenstern (FGM) Family

Let (X _i , Y _i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X _(r) denotes the rth-order statistic, then the Y's associated with X _(r) denoted by Y _[r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X _(r) and the entropy of its concomitants Y _[r]. In this family, we show that the mutual information between X _(r) and Y _[r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X _(r) and Y _[r] with the mutual information of (X _(r), Y _[r]) for the copula model of FGM family.     

Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.     

Reliability of stress–strength model for exponentiated Pareto distributions

In this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical exponentiated Pareto distributed random variables. Different point estimations and interval estimations are proposed. The point estimators obtained are maximum likelihood, uniformly minimum variance unbiased and Bayesian estimators. The interval estimations obtained are approximate, exact, bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval. Different methods and the corresponding confidence intervals are compared using Monte-carlo simulations.     

The Topp–Leone odd log-logistic family of distributions

We introduce a new class of continuous distributions named the Topp–Leone odd log-logistic family, which extends the one-parameter distribution pioneered by Topp and Leone [A family of J-shaped frequency functions. J Amer Statist Assoc. 1955;50:209–219]. We study some of its mathematical properties and describe two special cases. Further, we propose a regression model based on the new Topp–Leone odd log-logistic Weibull distribution. The usefulness and flexibility of the proposed family are illustrated by means of three real data sets.     

Topp–Leone generated family of distributions: Properties and applications

We introduce a new family of distributions based on a one-parameter distribution exhibiting bathtub-shaped hazard rates. We study the mathematical properties of the family and estimate its parameters by the method of maximum likelihood. Finally, the usefulness of the family is illustrated using a real dataset.     

New results on the Ristić–Balakrishnan family of distributions

ABSTRACT

Adding new shape parameters to expand a model into a larger family of distributions to provide significantly skewed and heavy-tails plays a fundamental role in distribution theory. For any continuous baseline G distribution, Risti? and Balakrishnan (2012 Risti?, M.M., Balakrishnan, N. (2012). The gamma exponentiated exponential distribution. J. Stat. Comput. Simul. 82:1191–1206.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) proposed the gamma-generated family of distributions with an extra positive shape parameter. They presented some special models of their family but did not study its properties. This paper examines some general mathematical properties of this family which hold for any baseline model. Some distributions are studied and a number of existing results in the literature can be recovered as special cases. We estimate the model parameters by maximum likelihood and illustrate the importance of the family by means of an application to a real data set.     

Parametric bootstrap goodness-of-fit testing for Wehrly–Johnson bivariate circular distributions

The Wehrly–Johnson family of bivariate circular distributions is by far the most general one currently available for modelling data on the torus. It allows complete freedom in the specification of the marginal circular densities as well as the binding circular density which regulates any dependence that might exist between them. We propose a parametric bootstrap approach for testing the goodness-of-fit of Wehrly–Johnson distributions when the forms of their marginal and binding densities are assumed known. The approach admits the use of any test for toroidal uniformity, and we consider versions of it incorporating three such tests. Simulation is used to illustrate the operating characteristics of the approach when the underlying distribution is assumed to be bivariate wrapped Cauchy. An analysis of wind direction data recorded at a Texan weather station illustrates the use of the proposed goodness-of-fit testing procedure.     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

Abstract

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.     

R–estimation of normed bivariate density functions

Normed bivariate density funtions were introduced by HOEFFDING (1940/41). In the present paper estimators for normed bivariate ranks and on a FOURIER series expansion in LEGENDRE polynomials. The estimation of normed bivarate density functions under positive dependence is also described     

Comparison of several Birnbaum–Saunders distributions

Birnbaum–Saunders (BS) distribution is widely used in reliability applications to model failure times. For several samples from possible different BS distributions, to prevent wrong conclusions in any further analysis, it is of importance to accompany a formal comparison for characteristic quantities of the distributions, including mean, quantile and reliability function difference. To this end, two test statistics, which are respectively based on the exact generalized p-value approach and the Delta method, are proposed and their behaviours are investigated. Simulation studies are carried out to examine the size and power performance of the newly proposed statistics. An interesting phenomenon is that in the finite sample simulations we conduct, the Delta method-based test almost uniformly outperforms the generalized p-value-based test although its sampling null distribution is simulated by Monte Carlo method. This might suggest that the sampling null distribution of the Delta method-based test statistic would have a fast convergence to its limit. The tests are also applied to analyse a real example on the fatigue life of 6061-T6 aluminium coupons for illustration.     

On a nonlinear Birnbaum–Saunders model based on a bivariate construction and its characteristics

Abstract

The Birnbaum-Saunders (BS) distribution is an asymmetric probability model that is receiving considerable attention. In this article, we propose a methodology based on a new class of BS models generated from the Student-t distribution. We obtain a recurrence relationship for a BS distribution based on a nonlinear skew–t distribution. Model parameters estimators are obtained by means of the maximum likelihood method, which are evaluated by Monte Carlo simulations. We illustrate the obtained results by analyzing two real data sets. These data analyses allow the adequacy of the proposed model to be shown and discussed by applying model selection tools.     

Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions

In an attempt to produce more realistic stress–strength models, this article considers the estimation of stress–strength reliability in a multi-component system with non-identical component strengths based on upper record values from the family of Kumaraswamy generalized distributions. The maximum likelihood estimator of the reliability, its asymptotic distribution and asymptotic confidence intervals are constructed. Bayes estimates under symmetric squared error loss function using conjugate prior distributions are computed and corresponding highest probability density credible intervals are also constructed. In Bayesian estimation, Lindley approximation and the Markov Chain Monte Carlo method are employed due to lack of explicit forms. For the first time using records, the uniformly minimum variance unbiased estimator and the closed form of Bayes estimator using conjugate and non-informative priors are derived for a common and known shape parameter of the stress and strength variates distributions. Comparisons of the performance of the estimators are carried out using Monte Carlo simulations, the mean squared error, bias and coverage probabilities. Finally, a demonstration is presented on how the proposed model may be utilized in materials science and engineering with the analysis of high-strength steel fatigue life data.     

The mixture of left–right truncated normal distributions

This paper introduces the mixture of left–right truncated normal distributions, from the spreads between bid and ask prices, as a statistical model for handle non-normality of asset price returns. It has been proved that there is only one maximum for the likelihood function of the new model.