Gaussian graphical models (GGMs) are well-established tools for probabilistic exploration of dependence structures using precision (inverse covariance) matrices. We propose a Bayesian method for estimating the precision matrix in GGMs. The method leads to a sparse and adaptively shrunk estimator of the precision matrix, and thus conduct model selection and estimation simultaneously. We extend this method in a regression setup with the presence of covariates. We consider both the linear as well as the nonlinear regressions in this GGM framework. Furthermore, to relax the assumption of the Gaussian distribution, we develop a quantile based approach for sparse estimation of graphs. We demonstrate that the resulting graph estimator is robust to outliers and applicable under general distributional assumptions. We discuss a few applications of the proposed models.

There are many situations where one expects an ordering among K>2 experimental groups or treatments. Although there is a large body of literature dealing with the analysis under order restrictions, surprisingly very little work has been done in the context of the design of experiments. Here, we provide some key observations and fundamental ideas which can be used as a guide for designing experiments when an ordering among the groups is known in advance. Designs maximizing power as well as designs based on single and multiple contrasts are discussed. The theoretical findings are supplemented by numerical illustrations.

Recurrence relations of moments which are useful to reduce the amount of direct computations quite considerably and usefully express the higher order moments of order statistics in terms of the lower order moments and hence make the evaluation of higher order moments easy. We have derived recurrence relations for single, double (product) and higher moments of various ordered random variables, like ordinary order statistics, progressively censored order statistics, generalized order statistics and dual generalized order statistics from some specific continuous distributions. It also deals with L-moments and TL-moments which are analogous of the ordinary moments. We have derived L-moments and TL-moments for some continuous distributions. These results have been applied to find the L-moment estimators and TL-moment estimators of the unknown parameters for some specific continuous distributions.

This work considers goodness-of-fit for the life test data with hybrid censoring. An alternative representation of the Kolmogorov-Smirnov (KS) statistics is provided under Type-I censoring. The alternative representation leads us to approximate the limiting distributions of the KS statistic as a functional of the Brownian bridge for Type-II, Type-I hybrid and Type-II hybrid censored data. The approximated distributions are used to obtain the critical values of the tests in this context.We found that the proposed KS test procedure for Type-II censoring has more power than the available one(s) in literature.

Let f be a step function on the circle with zero mean and rational discontinuities while alpha is a quadratic irrational. The point-wise ergodic theorem tells us that the ergodic sums, f(x)+f(x+alpha)+...+f(x+(n-1)alpha) is o(n) for almost every x but says nothing about its deviations from zero, that is, its discrepancy; the study of these deviations naturally draws us to the study of ergodic transformations on infinite measure spaces, viz., skew products over irrational rotations. In this talk, after a brief introduction to these terms, we will learn how the temporal statistics of the ergodic sums for x=0 can be studied via random affine transformations (RATs) leading to a central limit theorem and other fine properties like the visit times to a neighbourhood of 0 vis-a-vis bounded rational ergodicity (all of course time permitting). This is reporting on joint work with Jon Aaronson and Michael Bromberg.

We study the problem of discriminating Gaussian processes by analyzing the behavior of the underlying probability measures in an infinite-dimensional space. Motivated by singularity of a certain class of Gaussian measures, we first propose a data based transformation for the training data. For a J class classification problem, this transformation induces complete separation among the associated Gaussian processes. The misclassification probability of a component-wise classifier when applied on this transformed data asymptotically converges to zero. In finite samples, the empirical classifier is constructed and related theoretical properties are studied. This is a joint work with Juan A. Cuesta-Albertos.

In current automotive market, reliability and durability are important hygiene factors to the customer and customer’s perception about these parameters is established through length of product warranty period. Therefore, to be competitive in market, all companies aim to provide optimal warranty on their product. To decide optimal warranty period for a product, forecasting of cost to the companies due to particular warranty policy is essential. Automotive product warranty has two dimensions, time and mileage (t,m). Hence, to forecast warranty cost one has to model a bivariate distribution for (t,m) and project expected number of warranty returns within specified period (t,m). In this paper we bypass the bivariate distribution by using usage distribution and use multiple renewal processes induced by Weibull distributions to arrive at cost structure under various warranty policies.

We introduce standard mixture models and standard mixture designs as are well-known in the literature. Some of the less known models are also introduced briefly. Next we mention about known applications of mixture experiments in agriculture, food processing and pharmaceutical studies. Then we describe the framework of exact and approximate [or, continuous] mixture designs. A broad class of research problems posed and discussed in the published literature is presented with appropriate references.

We demonstrate that for discrete Markov dependent rvs, the normal approximation can be effectively replaced by compound Poisson approximation..In case of three state Markov chain, the effect of symmetry will be estimated.

Some convergence results in probability measures are used to prove approximations to the factorial of natural numbers and the Weierstrass Approximation Theorem.