Shrinkage Estimation for the Diagonal Multivariate Natural Exponential Families
Open Access
- Author:
- Siapoutis, Nikolas
- Graduate Program:
- Statistics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- May 31, 2022
- Committee Members:
- Donald Richards, Dissertation Advisor
Bharath Kumar Sriperumbudur, Chair & Dissertation Advisor
Alexei Novikov, Outside Unit & Field Member
Matthew Reimherr, Major Field Member
Ephraim Hanks, Professor in Charge/Director of Graduate Studies - Keywords:
- shrinkage estimators
diagonal multivariate natural exponential families
unbiased estimate of risk
semi-parametrics
sampling methods - Abstract:
- In this dissertation, we derive and study shrinkage estimators of the parameters of a high-dimensional diagonal natural exponential family of probability distributions. More broadly, we study shrinkage estimation of the parameters of distributions for which the diagonal entries of the covariance matrix are certain quadratic functions of the mean parameter. We propose two classes of semi-parametric shrinkage estimators for the mean of the population, and we construct unbiased estimators of the corresponding risk. We establish the asymptotic consistency and convergence rates for these shrinkage estimators under squared error loss as both $n$, the sample size, and $p$, the dimension, tend to infinity. Further, we specialize these results for the diagonal multivariate natural exponential families, which have been classified as consisting of the normal, Poisson, gamma, multinomial, negative multinomial, and hybrid classes of distributions, and we deduce consistency of our estimators. We deduce consistency of our estimators in the normal, gamma, and negative multinomial cases if $p n^{-1/3}(\log n)^{4/3} \rightarrow 0$ as $n,p \rightarrow \infty$, and for the Poisson and multinomial cases if $pn^{-1/2} \rightarrow 0$ as $n,p \rightarrow \infty$ To evaluate the performance of our mean shrinkage estimators, we carry out a simulation study for the multivariate gamma and multivariate Poisson classes of distributions. We begin by deriving the probability density functions of these two classes of distributions and establish some related regression properties. We propose several acceptance-rejection sampling algorithms and apply two versions of Metropolis algorithm to generate data from the multivariate gamma distribution, all-at-once and variable-at-a-time Metropolis algorithms. We propose reduction schemes for simulating observations from a multivariate Poisson distribution. We also approximate the probability density function of the multivariate Poisson distribution by applying the saddlepoint approximation method. Finally, we apply a variable-at-a-time Metropolis algorithm to generate data from the approximated probability density function. The simulation studies show the proposed estimators to achieve lower risk than the maximum likelihood estimator, thereby demonstrating the superiority of the proposed shrinkage estimators over the maximum likelihood estimator.