Fixed Size Confidence Regions for Parameters of a Logistic Regression Model

Abstract

Let $(\mathbf{X}_i, Y_i)$ be independent, identically distributed observations that satisfy a logistic regression model; that is, for each $i, \log \lbrack P(Y_i = 1 | \mathbf{X}_i)/P(Y_i = 0 |\mathbf{X}_i)\rbrack = \mathbf{X}^T_i \beta_0$, where $Y_i \in \{0, 1\}, \mathbf{X}_i \in \mathbf{R}^p$ and $\beta_0 \in \mathbf{B}^p$ is the unknown parameter vector of the model. The marginal distribution of the covariate vectors $\mathbf{X}_i$ is assumed to be unknown. Sequential procedures for constructing fixed size and fixed proportional accuracy confidence regions for $\beta_0$ are proposed and shown to be asymptotically efficient as the size of the region becomes small.