Inadmissibility of Studentized Tests for Normal Order Restricted Models

Abstract

Consider the model where $X_{ij}, i = 1,\ldots, k; j = 1,2,\ldots, n_i; n_i \geq 2$, are observed. Here $X_{ij}$ are independent $N(\theta_i,\sigma^2), \theta_i, \sigma^2$ unknown. Let $X_i = \sum^n_{j = 1}X_{ij}/n_i, \mathbf{X}' = (X_1,\ldots, X_k), \mathbf{\theta}' = (\theta_1,\ldots,\theta_k), V = \sum^k_{i = 1} \sum^{n_i}_{j = 1}X^2_{ij} - n \sum^k_{i = 1}X^2_i$. Let $\mathbf{A}_1$ be a $(k - m) \times k$ matrix of rank $(k - m) \geq 2$ and test $H: \mathbf{A}_1\mathbf{\theta} = \mathbf{0}$ versus $K - H$ where $K: \mathbf{A}_1\mathbf{\theta} \geq \mathbf{0}$. Suppose we assume $\sigma^2$ known and consider a constant size $\alpha$ test $(\alpha < 1/2)$ which is admissible for $H$ versus $K - H$ based on $\mathbf{X}$. Next assume $\sigma^2$ is unknown. Consider the same test but now as a function of $\mathbf{X}/V^{1/2}$ (i.e., Studentize the test). The resulting test is inadmissible. Examples are noted.