Trace Class Markov Chains for Bayesian Inference with Generalized Double Pareto Shrinkage Priors

Bayesian shrinkage methods have generated a lot of interest in recent years, especially in the context of high‐dimensional linear regression. In recent work, a Bayesian shrinkage approach using generalized double Pareto priors has been proposed. Several useful properties of this approach, including the derivation of a tractable three‐block Gibbs sampler to sample from the resulting posterior density, have been established. We show that the Markov operator corresponding to this three‐block Gibbs sampler is not Hilbert–Schmidt. We propose a simpler two‐block Gibbs sampler and show that the corresponding Markov operator is trace class (and hence Hilbert–Schmidt). Establishing the trace class property for the proposed two‐block Gibbs sampler has several useful consequences. Firstly, it implies that the corresponding Markov chain is geometrically ergodic, thereby implying the existence of a Markov chain central limit theorem, which in turn enables computation of asymptotic standard errors for Markov chain‐based estimates of posterior quantities. Secondly, because the proposed Gibbs sampler uses two blocks, standard recipes in the literature can be used to construct a sandwich Markov chain (by inserting an appropriate extra step) to gain further efficiency and to achieve faster convergence. The trace class property for the two‐block sampler implies that the corresponding sandwich Markov chain is also trace class and thereby geometrically ergodic. Finally, it also guarantees that all eigenvalues of the sandwich chain are dominated by the corresponding eigenvalues of the Gibbs sampling chain (with at least one strict domination). Our results demonstrate that a minor change in the structure of a Markov chain can lead to fundamental changes in its theoretical properties. We illustrate the improvement in efficiency resulting from our proposed Markov chains using simulated and real examples.     

Additive regression model for stationary and ergodic continuous time processes

The main purpose of the present work is to introduce and investigate a simple kernel procedure based on marginal integration that estimates the regression function for stationary and ergodic continuous time processes in the setting of the additive model introduced by Stone (1985 Stone, C.J. (1985). Additive regression and other nonparametric models. Ann. Stat. 13(2):689–705.[Crossref], [Web of Science ®] , [Google Scholar]). We obtain the uniform almost sure consistency with exact rate and the asymptotic normality of the kernel-type estimators of the components of the additive model. Asymptotic properties of these estimators are obtained, under mild conditions, by means of martingale approaches. Finally, a general notion of the bootstrapped additive components, constructed by exchangeably weighting sample, is presented.     

Discrete distributions in the extended FGM family: The p.g.f. approach

In this article the probability generating functions of the extended Farlie–Gumbel–Morgenstern family for discrete distributions are derived. Using the probability generating function approach various properties are examined, the expressions for probabilities, moments, and the form of the conditional distributions are obtained. Bivariate version of the geometric and Poisson distributions are used as illustrative examples. Their covariance structure and estimation of parameters for a data set are briefly discussed. A new copula is also introduced.     

Markov switching model of nonlinear autoregressive with skew-symmetric innovations

We consider data generating structures which can be represented as a Markov switching of nonlinear autoregressive model with considering skew-symmetric innovations such that switching between the states is controlled by a hidden Markov chain. We propose semi-parametric estimators for the nonlinear functions of the proposed model based on a maximum likelihood (ML) approach and study sufficient conditions for geometric ergodicity of the process. Also, an Expectation-Maximization type optimization for obtaining the ML estimators are presented. A simulation study and a real world application are also performed to illustrate and evaluate the proposed methodology.     

DENSITY FUNCTION APPROXIMATION WITH PARTICULAR REFERENCE TO THE DISTRIBUTION OF OVERLAP OF CIRCLES

Given that two circles overlap, the area in common is a function of the distance between their centres. This paper adopts a suitable random distribution for the intercentre distance and then derives the distribution of the area of overlap. An approximation is sought for the density function using a criterion which enables bounds to be placed on the difference between the moments of the density function and those of the approximation. This is an approach of general applicability. The importance of matching the end-point behaviour of the density and the approximation is stressed. It is shown that the distribution of the area of overlap may be well approximated by a mixture of beta distributions in which the parameters change smoothly with the ratio of radii.     

An autoregressive process with geometric α-laplace marginals

Geometric Laplace and geometric α-Laplace distributions are studied and certain limit properties are derived. An autoregressive process with geometric Laplace stationary marginal distribution is introduced and its properties are studied. These results are generalized to geometric α-Laplace case also and applications are discussed.     

A Review on Characterizations of Discrete Distributions Based on Records and kth Records

This is a survey of characterizations of discrete distributions via properties of record values. Characterization results based on records and weak records are presented. Then the concepts of kth records, strong kth records, and weak kth records are recalled. Finally, characterizations of the geometric parent involving these three types of kth records are discussed.     

Bayes Estimation of Shift Point in Geometric Sequence

A sequence of independent lifetimes X ₁, X ₂,…, X _m , X _m+1,… X _n were observed from geometric population with parameter q ₁ but later it was found that there was a change in the system at some point of time m and it is reflected in the sequence after X _m by change in parameter q ₂. The Bayes estimates of m, q ₁, q ₂, reliability R ₁ (t) and R ₂ (t) at time t are derived for symmetric and asymmetric loss functions under informative and non informative priors. A simulation study is carried out.     

On Some New α-Modified Binomial and Poisson Distributions and Their Applications

A new class of α-modified binomial distribution has been proposed, and its distributional properties like probability generating function (pgf), moments, and their interrelations have been studied. Two new α-modified Poisson distributions and Poisson distribution have been obtained as limiting distributions. Modified binomial and Poisson distributions introduced by Berg and Jaworski (1988 Berg , S. , Jaworski , J. ( 1988 ). Modified binomial and Poisson distributions with application in random mapping theory . J. Statist. Plann. Infer. 18 : 313 – 322 . [Google Scholar]) have been seen as particular cases. Mixture distributions of α-modified binomial distributions have been derived. A new distributions called α-modified binomial distributions of type j, their moment properties, limiting distributions as α-modified Poisson distribution of type j, their different convolution properties, pgf, parameter estimators have been studied. Two more new distributions namely Doubly α-modified binomial distributions of type (i, j) and α-modified weighted generalized Poisson distributions of type (j ? 1) have also been studied. Various α-modified binomial and Poisson distributions of Berg and Mutafchiev (1990 Berg , S. , Mutafchiev , L. ( 1990 ). Random mapping with an attracting center: Lagrangian distributions and a regression function . J. Appl. Probab. 27 : 622 – 636 . [Google Scholar]) and Berg and Nowicki (1991 Berg , S. , Nowicki , K. ( 1991 ). Statistical inference for a class of modified power series distributions with applications to random mapping theory . J. Statist. Plann. Infer. 28 : 247 – 261 . [Google Scholar]) have been seen as special cases. Application of some of these proposed distributions have been identified.     

EWMA Control Charts for Monitoring High Yield Processes

The main objective of this article is to scrutinize the efficiency and verify the performance superiority of the one-sided EWMA control chart on high-yield processes. The proposed control chart is designed to detect both upward and downward shifts of the fraction of non conforming products and is developed based on non transformed geometric counts. Its algorithmic function is theoretically established and numerous performance measures are extracted using analytical methods based on the Markov modeling of the chart. Comparisons with traditional high yield control charts are conducted. Optimality tables and nomograms are included to help graphical determination of the optimal chart parameters.