Bivariate log Birnbaum–Saunders distribution

Univariate Birnbaum–Saunders distribution has received a considerable amount of attention in recent years. Rieck and Nedelman (A log-linear model for the Birnbaum–Saunders distribution. Technometrics, 1991;33:51–60) introduced a log Birnbaum–Saunders distribution. The main aim of this paper is to introduce bivariate log Birnbaum–Saunders distribution. The proposed model is symmetric and it has five parameters. It can be obtained using Gaussian copula. Different properties can be obtained using copula structure. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained explicitly. Two-dimensional profile likelihood approach may be adopted to compute the MLEs. We propose some alternative estimators also, which can be obtained quite conveniently. The analysis of one data set is performed for illustrative purposes. Finally, it is observed that this model can be used as a bivariate log-linear model, and its multivariate generalization is also quite straight forward.     

On a Bivariate Pólya-Aeppli Distribution

In this article, we use the bivariate Poisson distribution obtained by the trivariate reduction method and compound it with a geometric distribution to derive a bivariate Pólya-Aeppli distribution. We then discuss a number of properties of this distribution including the probability generating function, correlation structure, probability mass function, recursive relations, and conditional distributions. The generating function of the tail probabilities is also obtained. Moment estimation of the parameters is then discussed and illustrated with a numerical example.     

Bayesian Inference in Marshall–Olkin Bivariate Exponential Shared Gamma Frailty Regression Model under Random Censoring

Many analyses in the epidemiological and the prognostic studies and in the studies of event history data require methods that allow for unobserved covariates or “frailties”. We consider the shared frailty model in the framework of parametric proportional hazard model. There are certain assumptions about the distribution of frailty and baseline distribution. The exponential distribution is the commonly used distribution for analyzing lifetime data. In this paper, we consider shared gamma frailty model with bivariate exponential of Marshall and Olkin (1967 Marshall, A.W., Olkin, I. (1967). A multivariate exponential distribution. J. Am. Stat. Assoc. 62:30–44.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) distribution as baseline hazard for bivariate survival times. We solve the inferential problem in a Bayesian framework with the help of a comprehensive simulation study and real data example. We fit the model to the real-life bivariate survival data set of diabetic retinopathy data. We introduce Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in the proposed model and then compare the true values of the parameters with the estimated values for different sample sizes.     

Inference for IZAWA’S Bivariate Gamma Distribution

Abstract

In this article, we aim to establish some theoretical properties of Izawa’s bivariate gamma distribution having equal shape parameters. First, we propose a procedure to obtain the maximum likelihood estimates and derive an expression for the Fisher information. Simulation studies illuminate the properties of maximum likelihood estimators. We also establish an asymptotic test for independence based on the limiting distribution of maximum likelihood estimators.     

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.     

Test of Independence for Baker’s Bivariate Distributions

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. Journal of Multivariate Analysis 99: 2312–2327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new method for constructing multivariate distributions with given marginals based on order statistics. In this paper, we provide a test of independence for a pair of absolutely continuous random variables (X, Y) jointly distributed according to Baker’s bivariate distributions. Our purpose is to test the hypothesis that X and Y are independent versus the alternative that X and Y are positively (negatively) quadrant dependent. The asymptotic distribution of the proposed test statistic is investigated. Also, the powers of the proposed test and the class of distribution-free tests proposed by Kochar and Gupta (1987 Kochar, S. G., Gupta, R. P. (1987). Competitors of Kendall-tau test for testing independence against positive quadrant dependence. Biometrika 74(3): 664–666.[Crossref], [Web of Science ®] , [Google Scholar]) are compared empirically via a simulation study.     

Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

We consider non-parametric estimation of a continuous cdf of a random vector (X ₁, X ₂). With bivariate RC data, it is stated in van der Laan (1996 Van der Laan , M. J. ( 1996 ) Efficient estimation in the bivariate censoring model and repairing NPMLE . Ann. Statist. 24 : 596 – 627 .[Crossref], [Web of Science ®] , [Google Scholar], p. 598¹⁰, Ann. Statist.), Quale et al. (2006 Quale , C. M. , van der Laan , M. J. , Robins , J. R. ( 2006 ). Locally efficient estimation with bivariate right-censored data . JASA. 101 : 1076 – 1084 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar])).” The claim is based on a result in Tsai et al. (1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], p.1352, Ann. Statist.) that if X ₁ is right censored but not X ₂, then common ways for defining one NPMLE lead to inconsistency. If X ₁ is right censored and X ₂ is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986 Tsai , W. Y. , Leurgans , S. , Crowley , J. ( 1986 ). Nonparametric estimation of a bivariate survival function in the presence of censoring . Ann. Statist. 14 : 1351 – 1365 .[Crossref], [Web of Science ®] , [Google Scholar], Ann. Statist.).     

SUB-SAMPLE MANN–WHITNEY–WILCOXON TEST

ABSTRACT

This paper presents a class of sub-sample rank-sum statistics to test the stochastic ordering between two distributions. The proposed class includes as special case the min–max test of Öztürk.[1] Öztürk, Ö. 2001. A Generalization of Ahmad's Class of Mann–Whitney–Wilcoxon Statistics. Austrl. and New Zealand J. Statist., 43: 67–74.  [Google Scholar] It is shown that the asymptotic distribution of the test statistic is normal and its Pitman asymptotic efficiency is as good as or higher than the competitors in the class of sub-sample Mann–Whitney–Wilcoxon statistics.     

Inverse Circular–Linear/Linear–Circular Regression

We propose two distance-based methods and two likelihood-based methods of inversely regressing a linear predictor on a circular variable, and of inversely regressing a circular predictor on a linear variable. An asymptotic result on least circular distance estimators is provided in the Appendix. We present likelihood-based methods for symmetrical and asymmetrical errors in each situation. The utility of our methodology in each situation is illustrated by applying it to real data sets in engineering and environmental science. We then compare their performances using a cross validation method.     

Marshall–Olkin gamma–Weibull distribution with applications

Abstract

A Marshall–Olkin variant of the Provost type gamma–Weibull probability distribution is being introduced in this paper. Some of its statistical functions and numerical characteristics among others characteristics function, moment generalizing function, central moments of real order are derived in the computational series expansion form and various illustrative special cases are discussed. This density function is utilized to model two real data sets. The new distribution provides a better fit than related distributions as measured by the Anderson–Darling and Cramér–von Mises statistics. The proposed distribution could find applications for instance in the physical and biological sciences, hydrology, medicine, meteorology, engineering, etc.     

Additional Tables for Steel–Dwass–Critchlow–Fligner Distribution-Free Multiple Comparisons of Three Treatments

ABSTRACT

Additional critical points are presented for the Steel–Dwass–Critchlow–Fligner distribution-free multiple comparison procedure for comparing all pairs of three population medians in the one-way layout. A computational technique developed by van de Wiel is used to find critical points yielding an experimentwise error rate of approximately 0.01, 0.05, and 0.10 for a total sample size of at most 30, with individual sample sizes from 4 to 10 and a maximum sample size of at least 8, and for equal sample sizes from 8 to 14. Additional discussion is given regarding step-down testing methods and the dangers of using the Steel–Dwass–Critchlow–Fligner procedure with unequal sample sizes if two of the sample sizes are very small.     

Approximating the Conway–Maxwell–Poisson distribution normalization constant

By adding a second parameter, Conway and Maxwell created a new distribution for situations where data deviate from the standard Poisson distribution. This new distribution contains a normalization constant expressed as an infinite sum whose summation has no known closed-form expression. Shmueli et al. produced an approximation for this sum but proved that it was valid only for integer values of the second parameter, although they conjectured it was also valid for non-integers. Here we prove their conjecture to be true and discuss for what range of parameters the approximation can be accurately applied.     

Marshall–Olkin q-Weibull distribution and max–min processes

In this paper, we introduce a new probability model known as Marshall–Olkin q-Weibull distribution. Various properties of the distribution and hazard rate functions are considered. The distribution is applied to model a biostatistical data. The corresponding time series models are developed to illustrate its application in times series modeling. We also develop different types of autoregressive processes with minification structure and max–min structure which can be applied to a rich variety of contexts in real life. Sample path properties are examined and generalization to higher orders are also made. The model is applied to a time series data on daily discharge of Neyyar river in Kerala, India.     

Bias–variance and breadth–depth tradeoffs in respondent-driven sampling

Respondent-driven sampling (RDS) is a link-tracing network sampling strategy for collecting data from hard-to-reach populations, such as injection drug users or individuals at high risk of being infected with HIV. The mechanism is to find initial participants (seeds), and give each of them a fixed number of coupons allowing them to recruit people they know from the population of interest, with a mutual financial incentive. The new participants are again given coupons and the process repeats. Currently, the standard RDS estimator used in practice is known as the Volz–Heckathorn (VH) estimator. It relies on strong assumptions about the underlying social network and the RDS process. Via simulation, we study the relative performance of the plain mean and VH estimators when assumptions of the latter are not satisfied, under different network types (including homophily and rich-get-richer networks), participant referral patterns, and varying number of coupons. The analysis demonstrates that the plain mean outperforms the VH estimator in many but not all of the simulated settings, including homophily networks. Also, we highlight the implications of multiple recruitment and varying referral patterns on the depth of RDS process. We develop interactive visualizations of the findings and RDS process to further build insight into the various factors contributing to the performance of current RDS estimation techniques.     

Smoothed Mann–Whitney–Wilcoxon Procedure for Two-Sample Location Problem

This study is mainly concerned with estimating a shift parameter in the two-sample location problem. The proposed Smoothed Mann–Whitney–Wilcoxon method smooths the empirical distribution functions of each sample by using convolution technique, and it replaces unknown distribution functions F(x) and G(x ? Δ₀) with the new smoothed distribution functions F _s (x) and G _s (x ? Δ₀), respectively. The unknown shift parameter Δ₀ is estimated by solving the gradient function S _n (Δ) with respect to an arbitrary variable Δ. The asymptotic properties of the new estimator are established under some conditions that are similar to the Generalized Wilcoxon procedure proposed by Anderson and Hettmansperger (1996 Anderson , G. F. , Hettmansperger , T. P. ( 1996 ). Generalized Wilcoxon methods for the one and two-sample location models . In: Brunner , E. , Denker , M. , eds. Research Developments in Probability and Statistics: Festschrift in Honor of Madan L. Puri on the Occasion of his 65th Birthday . Zeist, The Netherlands : VSP BV , pp. 303 – 317 . [Google Scholar]). Some of these properties are asymptotic normality, asymptotic level confidence interval, and hypothesis testing for Δ₀. Asymptotic relative efficiency of the proposed method with respect to the least squares, Generalized Wilcoxon and Hodges and Lehmann (1963 Hodges , J. L. , Lehmann , E. L. ( 1963 ). Estimates of location based on rank tests . Ann. Mathemat. Statist. 34 : 598 – 611 .[Crossref] , [Google Scholar]) procedures are also calculated under the contaminated normal model.     

Erlang–Lindley distribution

In this paper, Erlang–Lindley distribution (ErLD) is proposed which offers a more flexible model for waiting time data. It has the property that it can accommodate increasing, bathtub, and inverted bathtub shapes. Several statistical and reliability properties are derived and studied. The moments, its associated measures, and the limiting distributions of order statistics are derived. The model parameters are estimated by maximum likelihood and method of moments. An application of the proposed distribution to some waiting time data shows that it can give a better fit than other important lifetime models.     

Reliability characteristics of Farlie–Gumbel–Morgenstern family of bivariate distributions

Abstract

In this paper, we study the Farlie–Gumbel–Morgenstern family of bivariate distributions from a reliability point of view. The properties of this family of distributions and the association between the two variables are investigated by studying the local dependence function and the association measure defined by Clayton (1978 Clayton, D.G. (1978). A model for association in bivariate life tables and its applications in epidemiological studies of familial tendency in chronic disease incidence. Biometrika 65:141–151.[Crossref], [Web of Science ®] , [Google Scholar]). We also study the effect of the association parameter on the hazard components, the failure rate of the series system, and the regression mean residual life of a parallel system. Stochastic comparisons with respect to the association parameter are also studied. Some examples are provided to illustrate the results.