T-Inverse Exponential Family Of Distributions

Recently, several methods have been introduced to generate neoteric distributions with more exibility, like T-X, T-R [Y] and alpha power. The T-Inverse exponential [Y] neoteric family of distributons is proposed in this paper utilising the T-R [Y] method. A generalised inverse exponential (IE) distribution family has been established. The distribution family is generated using quantile functions of some dierent distributions. A number of general features in the T-IE [Y] family are examined, like mean deviation, mode, moments, quantile function, and entropies. A special model of the T-IE [Y] distribution family was one of those old distributions. Certain distribution examples are produced by the T-IE [Y] family. An applied case was presented which showed the importance of the neoteric family.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

Inference on Positive Exponential Family of Distributions (PEFD) through Transformation Method

The estimation of R(t) and R = Pr(Y > X) for the Positive Exponential Family of Distribution(PEFD) is considered. The UMVUE's, MLE's and Confidence Interval are derived.These estimators are derived through the method of Transformation. The α = P(X > γ),which is termed as probability of disaster is also derived when random stress X follows PEFDand finite strength follows Power function distribution.

Approximation of Information Divergences for Statistical Learning with Applications

In this paper we give a partial response to one of the most important statistical questions, namely, what optimal statistical decisions are and how they are related to (statistical) information theory. We exemplify the necessity of understanding the structure of information divergences and their approximations, which may in particular be understood through deconvolution. Deconvolution of information divergences is illustrated in the exponential family of distributions, leading to the optimal tests in the Bahadur sense. We provide a new approximation of I-divergences using the Fourier transformation, saddle point approximation, and uniform convergence of the Euler polygons. Uniform approximation of deconvoluted parts of I-divergences is also discussed. Our approach is illustrated on a real data example.

Flexible Bayesian Models for Inferences From Coarsened, Group-Level Achievement Data

This article proposes a flexible extension of the Fay–Herriot model for making inferences from coarsened, group-level achievement data, for example, school-level data consisting of numbers of students falling into various ordinal performance categories. The model builds on the heteroskedastic ordered probit (HETOP) framework advocated by Reardon, Shear, Castellano, and Ho by allowing group parameters to be modeled with regressions on group-level covariates, and residuals modeled using the flexible exponential family of distributions recommended by Efron. We demonstrate that the alternative modeling framework, termed the “Fay–Herriot heteroskedastic ordered probit” (FH-HETOP) model, is useful for mitigating some of the challenges with direct maximum likelihood estimators from the HETOP model. We conduct a simulation study to compare the costs and benefits of several methods for using the FH-HETOP model to estimate group parameters and functions of them, including posterior means, constrained Bayes estimators, and the “triple goal” estimators of Shen and Louis. We also provide an application of the FH-HETOP model to math proficiency data from the Early Childhood Longitudinal Study. Code for estimating the FH-HETOP model and conducting supporting calculations is provided in a new package for the R environment.