On Sharp Jensen's Inequality and Some Unusual Applications

The purpose of this article is two-fold. First, we find it very interesting to explore a kind of notion of optimality of the customary Jensen-bound among all Jensen-type bounds. Without this result, the customary Jensen-bound stood alone simply as just another bound. The proposed notion and the associated optimality are important given that in some situations the Jensen's inequality does leave us empty handed.

When it comes to highlighting Jensen's inequality, unfortunately only a handful of nearly routine applications continues to recycle time after time. Such encounters rarely produce any excitement. This article may change that outlook given its second underlying purpose, which is to introduce a variety of unusual applications of Jensen's inequality. The collection of our important and useful applications and their derivations are new.     

On the Asymmetric Marcinkiewicz-Zygmund Strong Law of Large Numbers for Linear Random Fields

In this article, the asymmetric Marcinkiewicz-Zygmund strong law of large numbers for linear random field under negative association is obtained. Our result generalizes a result in Gut and Studtmüller (2009 Gut , A. , Studtmüller , U. ( 2009 ) An asymmetric Marcinkiewicz-Zygmund LLN for random fields . Statist. Probab. Lett. 79 : 1016 – 1020 .[Crossref], [Web of Science ®] , [Google Scholar]). An asymmetric Marcinkiewicz-Zygmund LLN for random fields to the linear random field by using the Beverige-Nelson decomposition.     

Flexible Bivariate INAR(1) Processes Using Copulas

Multivariate count time series data occur in many different disciplines. The class of INteger-valued AutoRegressive (INAR) processes has the great advantage to consider explicitly both the discreteness and autocorrelation characterizing this type of data. Moreover, extensions of the simple INAR(1) model to the multi-dimensional space make it possible to model more than one series simultaneously. However, existing models do not offer great flexibility for dependence modelling, allowing only for positive correlation. In this work, we consider a bivariate INAR(1) (BINAR(1)) process where cross-correlation is introduced through the use of copulas for the specification of the joint distribution of the innovations. We mainly emphasize on the parametric case that arises under the assumption of Poisson marginals. Other marginal distributions are also considered. A short application on a bivariate financial count series illustrates the model.     

On the Bayes estimators of the parameters of size-biased generalized power series distributions

ABSTRACT

In this paper, we derive the Bayes estimators of functions of parameters of the size-biased generalized power series distribution under squared error loss function and weighted square error loss function. The results of size-biased GPSD are then used to obtain particular cases of the size-biased negative binomial, size-biased logarithmic series, and size-biased Poisson distributions. These estimators are better than the classical minimum variance unbiased estimators in the sense that they increase the range of the estimation. Finally, an example is provided to illustrate the results and a goodness of fit test is done using the maximum likelihood and Bayes estimators.     

Bayesian credibility limits for small finite population proportions:stratified and cluster sampling

Bayes credibility limits for small proportions from stratified and fixed size cluster samples are discussed. Ericson’s (JRSS B (1969)) Beta Binomial and Dirichlet-Multinomial priors are used. Approximate limits that are appropriate for large samples and small proportions are derived in both cases. These allow asymptotic comparisons of the efficacy of stratified and cluster sampling relative to simple random sampling for estimating small proportions. Procedures for the selection of hyper parameters are also presented.     

Multi-stage estimation of the common location parameter of several exponential distributions

The problem of constructing a confidence interval of ‘preassigned width and coverage probability’ considered by Costanza/ Hamdy and Son(1986) is further analyzed. Several multi-stage estimation procedures [ like, purely sequential, accelerated sequential and three-stage procedures ] are utilized to deal with the same estimation problem. The relative advantages and disadvantages of these procedures are discussed.     

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.     

Flexible Bivariate Count Data Regression Models

The article develops a semiparametric estimation method for the bivariate count data regression model. We develop a series expansion approach in which dependence between count variables is introduced by means of stochastically related unobserved heterogeneity components, and in which, unlike existing commonly used models, positive as well as negative correlations are allowed. Extensions that accommodate excess zeros, censored data, and multivariate generalizations are also given. Monte Carlo experiments and an empirical application to tobacco use confirms that the model performs well relative to existing bivariate models, in terms of various statistical criteria and in capturing the range of correlation among dependent variables. This article has supplementary materials online.     

Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques?

Modeling data that are non-normally distributed with random effects is the major challenge in analyzing binomial data in split-plot designs. Seven methods for analyzing such data using mixed, generalized linear, or generalized linear mixed models are compared for the size and power of the tests. This study shows that analyzing random effects properly is more important than adjusting the analysis for non-normality. Methods based on mixed and generalized linear mixed models hold Type I error rates better than generalized linear models. Mixed model methods tend to have higher power than generalized linear mixed models when the sample size is small.