On prediction and estimation problems for some multivariate distributions

Abstract : The thesis addresses three distinct decision problems concerning prediction and estimation for multivariate distributions. (1) Predictive density estimators with integrated $L_1$ loss for spherically symmetric distributions: we extend the scale expansion improvements under integrated $L_1$ loss, derived for the univariate case by Kubokawa et al. (2017), to multivariate scenarios. We also provide scale expansion improvements on plug-in densities of the form $q(\\|y-\\hat{\\theta}(X)\\|^2)$ for cases where $\\theta$ is restricted to a compact parameter space, even when $\\hat{\\theta}(X)$ is adapted to the parameter space. The findings also encompass a broader class of loss functions of the form $\\gamma(L_1(\\theta,\\hat{q}))$ with strictly increasing $\\gamma$. (2) Bayesian inference and prediction for mean-mixtures of normal distributions: we explore the problem of predictive density estimation for mean-mixtures of multivariate normal (MMN) distributions under Kullback-Leibler loss. We identify classes of plug-in type predictive densities and of Bayes predictive densities which are minimax and dominate the benchmark minimum equivariant estimator (MRE) for the case when the dimension of the location parameter is greater than or equal to four. Additionally, we present novel representations for Bayesian posterior distributions and predictive densities for MMN models, filling a gap in the existing literature. We also investigate implications for certain type of parametric restrictions on $\\theta$, and illustrate and comment the findings based on numerical evaluations. (3) Construction of proper Bayes minimax multiple shrinkage estimators: we address the canonical problem of estimating the mean of multivariate normal distributions under quadratic loss, and propose a feasible approach for constructing minimax pseudo Bayes multiple shrinkage estimators. This approach employs particular spherically symmetric priors, leading to scalable marginal densities, which satisfy Stein's minimaxity condition of superharmonicity. Furthermore, we demonstrate how the general framework allows for the construction of proper priors resulting in minimax multiple shrinkage estimators. Notably, we reveal the effectiveness of adjusted Strawderman-type priors in yielding proper Bayes minimax multiple shrinkage estimators.

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