Normal-Theory Approximations to Tests for Linear Hypotheses

Abstract

Let $\{\mathbf{y}_1, \cdots, \mathbf{y}_N\}$ be $N$ independent repetitions of an experiment for which $\mathscr{E}\mathbf{y} = \mathbf{X\beta}$ and $\mathscr{E}(\mathbf{y} - \mathbf{X\beta})(\mathbf{y} - \mathbf{X\beta} ' = \sigma^2\mathbf{I}$, where $\mathbf{X}$ is nonstochastic of full rank and $\mathbf{\beta'} = \lbrack \mathbf{\beta_1'}, \cdot, \mathbf{\beta}_r'\rbrack$ and $\sigma^2$ are unknown parameters. We investigate some large-sample properties of $N^{\frac{1}{2}} (\hat{\mathbf{\beta}}_N - \mathbf{\beta})$, of $V_N = N(\hat{\mathbf{\beta}}_N - \mathbf{\beta}_0)'\mathbf{T}^{-1} (\hat{\mathbf{\beta}}_N - \mathbf{\beta}_0)/\sigma^2$, and of $V_{Nj} = N(\hat{\mathbf{\beta}}_{Nj} - \mathbf{\beta}_{j0})'\mathbf{T}^{-1}_{jj} (\hat{\mathbf{\beta}}_{Nj} - \mathbf{\beta}_{j0})/\sigma^2;\quad 1 \leqq j \leqq r$ where $\hat{\mathbf{\beta}}_N = N^{-1} \mathbf{TX}'(\mathbf{y}_1 + \cdots + \mathbf{y}_N)$ and $\mathbf{T} = \lbrack \mathbf{T}_{ij}\rbrack = (\mathbf{X'X})^{-1}$. Our conclusions thus apply to problems of inference regarding $\mathbf{\beta}$ and $\{ \mathbf{\beta}_1,\cdots, \mathbf{\beta}_r\}$. Given certain higher-order moments of $\mathbf{y}$, we provide bounds of the distributions of $N^{\frac{1}{2}}(\hat{mathbf{\beta}}_N - \mathbf{\beta})$, of $V_n$, of $\{ V_{N1}, \cdots, V_{Nr}\}$, and of the variance is the sample variance. ratios $U_N = \sigma^2V_N/\hat{\sigma}_N^2$ and $\{U_{N1}, \cdots, U_{Nr}\}$, where $U_{Nj} = \sigma^2\mathbf{V}_{Nj}/\hat{\sigma}_N^2$ and $\hat{\sigma}_N^2$ is the sample variance.