Distribution of the Λ-Criterion for Sphericity Test and Its Connection to a Generalized Dirichlet Model

It is shown that the exact null distribution of the likelihood ratio criterion for sphericity test in the p-variate normal case and the marginal distribution of the first component of a (p ? 1)-variate generalized Dirichlet model with a given set of parameters are identical. The exact distribution of the likelihood ratio criterion so obtained has a general format for every p. A novel idea is introduced here through which the complicated exact null distribution of the sphericity test criterion in multivariate statistical analysis is converted into an easily tractable marginal density in a generalized Dirichlet model. It provides a direct and easiest method of computation of p-values. The computation of p-values and a table of critical points corresponding to p = 3 and 4 are also presented.     

Zero-Inflated Poisson Regression for Longitudinal Data

Medical and public health research often involve the analysis of repeated or longitudinal count data that exhibit excess zeros such as the number of yearly doctor visits by a group of individuals over a number of years. Zero-inflated Poisson (ZIP) regression models can be used to account for excess zeros in count data. We propose an extension of the ZIP model that is appropriate for longitudinal data. Our extension includes a non stationary, observation-driven time series model based correlation structure. We discuss estimation of the model parameters and the inefficiency of the estimators when the correlation structure is mis-specified. The model's application to the analysis of health care utilization data is also discussed.     

Determination of Single Sampling Plans by Attributes Under the Conditions of Zero-Inflated Poisson Distribution

When the manufacturing process is well monitored, occurrence of nondefects would be a frequent event in sampling inspection. The appropriate probability distribution of the number of defects is a zero-inflated Poisson (ZIP) distribution. In this article, determination of single sampling plans (SSPs) by attributes using unity values is considered, when the number of defects follows a ZIP distribution. The operating characteristic (OC) function of the sampling plan is derived. Plan parameters are obtained for some sets of values of (p₁, α, p₂, β). Numerical illustrations are given to describe the determination of SSP under ZIP distribution and to study its performance in comparison with Poisson SSP.     

Some Results for Beta Fréchet Distribution

Nadarajah and Gupta (2004 Nadarajah , S. , Gupta , A. K. ( 2004 ). The beta Fréchet distribution . Far East J. Theoret. Statist. 14 : 15 – 24 . [Google Scholar]) introduced the beta Fréchet (BF) distribution, which is a generalization of the exponentiated Fréchet (EF) and Fréchet distributions, and obtained the probability density and cumulative distribution functions. However, they did not investigate the moments and the order statistics. In this article, the BF density function and the density function of the order statistics are expressed as linear combinations of Fréchet density functions. This is important to obtain some mathematical properties of the BF distribution in terms of the corresponding properties of the Fréchet distribution. We derive explicit expansions for the ordinary moments and L-moments and obtain the order statistics and their moments. We also discuss maximum likelihood estimation and calculate the information matrix which was not given in the literature. The information matrix is numerically determined. The usefulness of the BF distribution is illustrated through two applications to real data sets.     

New Properties of the Kumaraswamy Distribution

The Kumaraswamy distribution is very similar to the Beta distribution but has the key advantage of a closed-form cumulative distribution function. This makes it much better suited than the Beta distribution for computation-intensive activities like simulation modeling and the estimation of models by simulation-based methods. However, in spite of the fact that the Kumaraswamy distribution was introduced in 1980, further theoretical research on the distribution was not developed until very recently (Garg, 2008 Garg , M. ( 2008 ). On distribution of order statistics from Kumaraswamy distribution . Kyunpook Mathemat. J. 48 : 411 – 417 .[Crossref] , [Google Scholar]; Jones, 2009 Jones , M. C. ( 2009 ). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages . Statist. Methodol. 6 ( 1 ): 70 – 81 .[Crossref] , [Google Scholar]; Mitnik, 2009 Mitnik , P. ( 2009 ). The Kumaraswamy distribution: a median-dispersion re-parameterization for regression modeling and simulation-based estimation. Available at SSRN: http://ssrn.com/abstract=1231587 . [Google Scholar]; Nadarajah, 2008 Nadarajah , S. ( 2008 ). On the distribution of Kumaraswamy . J. Hydrol. 348 : 568 – 569 .[Crossref], [Web of Science ®] , [Google Scholar]). This article contributes to this recent research and: (a) shows that Kumaraswamy variables exhibit closeness under exponentiation and under linear transformation; (b) derives an expression for the moments of the general form of the distribution; (c) specifies some of the distribution's limiting distributions; and (d) introduces an analytical expression for the mean absolute deviation around the median as a function of the parameters of the distribution, and establishes some bounds for this dispersion measure and for the variance.     

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.     

Non Parametric Estimation of Second-Order Diffusion Equation by Using Asymmetric Kernels

In this paper, we study the non parametric estimation of drift coefficient and diffusion coefficient in the second-order diffusion equation by using the asymmetric kernel functions, based on the difference of discrete time observations. The basic idea relies upon replacing the symmetric kernel by asymmetric kernel and provides a new way of obtaining the non parametric estimation for second-order diffusion equation. Under the appropriate assumptions, we prove that the proposed estimators of second-order diffusion equation are consistent and asymptotically follow normal distribution.     

Simultaneous tests for homogeneity of two zero-inflated (beta) populations

ABSTRACT

Motivated by an example in marine science, we use Fisher’s method to combine independent likelihood ratio tests (LRTs) and asymptotic independent score tests to assess the equivalence of two zero-inflated Beta populations (mixture distributions with three parameters). For each test, test statistics for the three individual parameters are combined into a single statistic to address the overall difference between the two populations. We also develop non parametric and semiparametric permutation-based tests for simultaneously comparing two or three features of unknown populations. Simulations show that the likelihood-based tests perform well for large sample sizes and that the statistics based on combining LRT statistics outperforms the ones based on combining score test statistics. The permutation-based tests have overall better performance in terms of both power and type I error rate. Our methods are easy to implement and computationally efficient, and can be expanded to more than two populations and to other multiple parameter families. The permutation tests are entirely generic and can be useful in various applications dealing with zero (or other) inflation.     

Local linear double and asymmetric kernel estimation of conditional quantiles

Abstract

In this work, we propose and investigate a family of non parametric quantile regression estimates. The proposed estimates combine local linear fitting and double kernel approaches. More precisely, we use a Beta kernel when covariate’s support is compact and Gamma kernel for left-bounded supports. Finite sample properties together with asymptotic behavior of the proposed estimators are presented. It is also shown that these estimates enjoy the property of having finite variance and resistance to sparse design.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.