| Microarray technology allows the simultaneous analysis of thousands of gene expressions from a single experiment. It has become a standard tool in molecular biology. Due to high experimental expenses, it is common that thousands of genes are measured with a small number of replications, which causes unstable variance estimation. Consequently, standard testing procedures have low power.;Many methods have been proposed to improve the variance estimation. The key idea is to borrow information across genes using shrinkage or Bayesian method. We consider optimal variance estimators by shrinking each gene-specific estimator toward the arithmetic mean based on original scale. The goal is to find the optimal shrinkage parameters under well-defined criteria such as the Stein and squared loss functions.;In Chapter 1, we briefly introduce the concepts of Baysian and shrinkage methods. Also we review the regularized t-test, the James-Stein shrinkage estimator and the optimal shrinkage estimator on the logarithmic scale which are the bases of our variance estimator. In Chapter 2, we derive the optimal shrinkage estimators for variances under the Stein and squared loss functions. We also consider the restriction for the shrinkage parameters. In Chapter 3, we derive the optimal shrinkage estimators for variance parameters raised to a nonzero power. It allows us to estimate directly power function of variances. In Chapter 4, we discuss how to estimate the optimal shrinkage estimators and present their asymptotic properties. In Chapter 5, we conduct simulations to evaluate the performance of the optimal shrinkage estimator and compare them with the existing estimators. The stability of variance estimators mainly depend on the number of replications for each gene. Our variance estimators are stable and consistent if the number of replications are not too small. From comparisons with existing methods, we conclude that the performance depends on the type of loss functions. While variance estimators shrinking toward the geometric mean outperform other estimators under the Stein loss function, the variance estimators shrinking toward the arithmetic mean outperform other estimators under the squared function. |
