Unbiased Testing in Exponential Family Regression

Abstract

Let $(X_{ij}, \mathbf{z}_i), i = 1, 2, \ldots, k, j = 1, 2, \ldots, n_i$, be independent observations such that $\mathbf{z}_i$ is a fixed $r \times 1$ vector $\lbrack r$ can be 0 (no $\mathbf{z}$'s observed) or $1, 2, \ldots, k - 1\rbrack$, and $X_{ij}$ is distributed according to a one-parameter exponential family which is log concave with natural parameter $\theta_i$. We test the hypothesis that $\theta = \mathbf{Z}\beta$, where $\theta = (\theta_1, \ldots, \theta_k)', \mathbf{Z}$ is the matrix whose $i$th row is $\mathbf{z}'_i$ and $\beta = (\beta_1, \ldots, \beta_r)'$ is a vector of parameters. We focus on $r = 2$ and $\mathbf{z}'_i = (1, z_i), i = 1, 2, \ldots, k, z_i < z_{i + 1}$. The null hypothesis on hand is thus of the form $\theta_i = \beta_1 + \beta_2z_i$. In such a case the model under the null hypothesis becomes logistic regression in the binomial case, Poisson regression in the Poisson case and linear regression in the normal case. We consider mostly the one-sided alternative that the second-order differences of the natural parameters are nonnegative. Such testing problems test goodness of fit vs. alternatives in which the natural parameters behave in a convex way. We find classes of tests that are unbiased and that lie in a complete class. We also note that every admissible test of constant size is unbiased. In some discrete situations we find the minimal complete class of unbiased admissible tests. Generalizations and examples are given.