On a Generalization of Bivariate Cauchy Distribution

This paper addresses a generalization of the bivariate Cauchy distribution discussed by Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]), derived from a trivariate normal distribution with a general correlation matrix. We obtain explicit expressions for the joint distribution function and joint density function, and show that they reduce in a special case to the corresponding expressions of Fang et al. (1990 Fang , K. T. , Kotz , S. , Ng , K. W. ( 1990 ). Symmetric Multivariate and Related Distributions . London : Chapman and Hall .[Crossref] , [Google Scholar]). Finally, we show that this generalized distribution is useful in determining the orthant probability of a bivariate skew-normal distribution of Azzalini and Dalla Valle (1996 Azzalini , A. , Dalla Valle , A. ( 1996 ). The multivariate skew-normal distribution . Biometrika 83 : 715 – 726 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Confidence Intervals for Linear Function of Variance Components in Two-Fold Nested Random Effects Model

In the article, we consider the unbalanced case of the two-way nested random effects model under partial balance. Using the method of generalized confidence intervals (GCIs) introduced in Weeranhandi (1993 Weeranhandi , S. ( 1993 ). Generalized confidence intervals . J. Amer. Statist. Assoc. 88 : 899 – 905 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar] 1995 Weeranhandi , S. ( 1995 ). Exact Statistical Methods for Data Analysis . New York : Springer-Verlag . [Google Scholar]), a new method is proposed for constructing confidence intervals on linear function of variance components. To compare the resulted intervals with the Modified Large Sample (MLS) intervals by Hernandez and Burdick (1993 Hernandez , R. P. , Burdick , R. K. ( 1993 ). Confidence intervals on the total variance in unbalanced two-fold nested designs . Biom. J. 35 : 515 – 522 .[Crossref] , [Google Scholar]), a simulation study is conducted. The results indicate that the proposed method performs better than the MLS method, especially for very unbalanced designs.     

Estimation of P(Y < X) for the Three-Parameter Generalized Exponential Distribution

This article considers the estimation of R = P(Y < X) when X and Y are distributed as two independent three-parameter generalized exponential (GE) random variables with different shape parameters but having the same location and scale parameters. A modified maximum likelihood method and a Bayesian technique are used to estimate R on the basis of independent complete samples. The Bayes estimator cannot be obtained in explicit form, and therefore it has been determined using an importance sampling procedure. An analysis of a real life data set is presented for illustrative purposes.     

Some Nonparametric Asymptotic Results for a Class of Stochastic Processes

This article provides a solution of a generalized eigenvalue problem for integrated processes of order 2 in a nonparametric framework. Our analysis focuses on a pair of random matrices related to such integrated process. The matrices are constructed considering some weight functions. Under asymptotic conditions on such weights, convergence results in distribution are obtained and the generalized eigenvalue problem is solved. Differential equations and stochastic calculus theory are used.     

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.).     

M-Estimator of a Generalized Linear Model with Measurement Errors

In this article, we propose a generalized linear model and estimate the unknown parameters using robust M-estimator. Under suitable conditions and by the strong law of large numbers and central limits theorem, the proposed M-estimators are proved to be consistent and asymptotically normal. We also evaluate the finite sample performance of our estimator through a Monte Carlo study.     

Estimation of Population Mean Using Known Coefficient of Variation of the Study Character in the Presence of Non Response

Using the known coefficient of variation of the study character, generalized and regression-type estimators for the population mean using two phase sampling in the presence of non response were proposed and their properties have been studied. The conditions under which the proposed estimators are more efficient than the relevant estimators have been obtained. The empirical studies were given in the support of the problems in the case of positive and negative correlation between the study and the auxiliary characters which show the increase in the efficiency of the proposed estimators using known coefficient of variation of the study character with respect to the relevant estimators.     

Schwarz Method for Penalized Quasi-Likelihood in Generalized Additive Models

The topic is penalized quasi-maximum likelihood estimation in generalized additive models by an approximation using a sequence of sub-models, here called blocks. The Schwarz method uses a sequence of sub-models, The technique might be useful to model comparison, where the fitted values from a sub-model are used as starting values for a larger model.

We show that the algorithm method converges in canonical link of generalized additive models, and a theorem about bound condition of sub-models convergence with uncanonical link.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.