Bayesian Estimation Under Shape Restriction and Some Deconvolution Problems

In this dissertation, we have explored Bayesian estimation under restrictions on the parameter space and deconvolution problems. The Bayesian estimation problem is motivated by two applications in hydrology and toxicology. The deconvolution part includes prediction of ordered random effects and recursive estimation of mixing distribution. In Bayesian estimation problem a family of monotone regression function is estimated. The functions start from zero and reach one and do not intersect in between. This problem is motivated by the depth duration curves in hydrology. We proposed a novel Bayesian solution based on Bernstein polynomial and established posterior consistency of our methodology using compactness of range and domain. A sufficient condition on the relative rate of increase of polynomial degree with respect to the number of observed points are explored. In other application, we modeled dose-response curves that arise in the context of toxicology. The problem of predicting order statistics has application is data masking and small area estimation problem. We have proposed a shrinkage type estimator and have derived the optimum value of the shrinkage parameter. The asymptotic value of the shrinkage coefficients depend on the Wasserstein distance between standardized distribution of observed and unobserved variable. In the process, we have also established a necessary and sufficient condition for a recent conjecture about the shrinkage coefficient to hold. The biggest advantage of this estimator is that it is distribution free. This makes the estimator robust and the estimator continues to perform well under model misspecification. Estimating the mixing density from a mixture is an important problem in statistics literature. This deconvolution can be carried out through a stochastic approximation based fast recursive algorithm. The main advantage of such algorithm is that it is extremely faster than the full Bayesian estimator. In current literature, properties have been shown under the independence of the observations. Here some dependent structures are explored. The recursive algorithm is also modified to address general mixing scenarios