New classes of discrete bivariate distributions with application to football data

Motivated by Lee and Cha [The American Statistician 69 (2015) 221–230], we introduce 18 new families of discrete bivariate distributions. We give explicit expressions for their probability mass functions, cumulative distribution functions, product moments, and moment-generating functions. We reanalyze the football data in Lee and Cha (2015 Lee, H., Cha, J.H. (2015). On two general classes of discrete bivariate distributions. Am. Stat. 69:221–230.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and show that some of the newly proposed distributions provide better fits than the two families proposed by Lee and Cha.     

Discrete distributions based on inter arrival times with application to football data

Motivated by McShane, Adrian, Bradlow and Fader [Journal of Business and Economic Statistics, 26, 2008, 369–378], we introduce thirteen discrete distributions. We give explicit expressions for their probability mass functions. We analyze two football data sets and show that some of the proposed distributions provide better fits than the distribution due to McShane, Adrian, Bradlow, and Fader.     

A note on Cochran test for homogeneity in one-way ANOVA and meta-analysis

In this paper we provide a formal yet simple and straightforward proof of the asymptotic χ² distribution for Cochran test statistic. Then, we show that the general form of this type of test statistics is invariant for the choice of weights. This fact is important since in practice many such test statistics are constructed with more complicated forms which usually require calculating generalized inverse matrices. Based on our results, we can simplify the construction of the test statistics. More importantly, properties such as anti-conservativeness of this type of test statistics can be drawn from Cochran test statistic. Furthermore, one can improve the performance of the tests by using some modified statistics with correction for small sample size situations.     

Some gamma distributions

Three new generalizations of the standard gamma distribution introduced by the author are reviewed. Various properties are derived for each distribution, including its hazard rate function and moments. An application is illustrated to drought data.     

Simple expressions for a bivariate chisquare distribution

The joint distribution of the estimated variances from a correlated bivariate normal distribution has a long history. However, its joint probability density function, conditional moments and product moments are only known as infinite series. In this paper, simpler expressions, mostly finite sums of elementary functions, are derived for these properties. Expressions are also derived for the joint moment generating function and the joint characteristic function.     

A note on the ratio of normal and Laplace random variables

The normal and Laplace are the two earliest known continuous distributions in statistics and the two most popular models for analyzing symmetric data. In this note, the exact distribution of the ratio | X / Y | is derived when X and Y are respectively normal and Laplace random variables distributed independently of each other. A MAPLE program is provided for computing the associated percentage points. An application of the derived distribution is provided to a discriminant problem.     

An alternative approach for compatibility of two discrete conditional distributions

ABSTRACT

Conditional specification of distributions is a developing area with increasing applications. In the finite discrete case, a variety of compatible conditions can be derived. In this paper, we propose an alternative approach to study the compatibility of two conditional probability distributions under the finite discrete setup. A technique based on rank-based criterion is shown to be particularly convenient for identifying compatible distributions corresponding to complete conditional specification including the case with zeros.The proposed methods are illustrated with several examples.     

The geometric exponential Poisson distribution

Many if not most lifetime distributions are motivated only by mathematical interest. Here, a new three-parameter distribution motivated mainly by lifetime issues is introduced. Some properties of the new distribution including estimation procedures, univariate generalizations and bivariate generalizations are derived. Two real data applications are described to show superior performance versus some known lifetime models.     

A new family of lifetime models

We introduce a new family of distributions by adding a parameter to the Marshall–Olkin family of distributions. Some properties of the new family of distributions are derived. A particular case of the family, a three-parameter generalization of the exponential distribution, is given special attention. The shape properties, moments, distributions of the order statistics, entropies and estimation procedures are derived. An application to a real data set is discussed.