Invariant Confidence Sequences for Some Parameters in a Multivariate Linear Regression Model

Abstract

Let $\mathbf{X}_1, \mathbf{X}_2,\cdots$ be independent $p$-variate normal vectors with $E \mathbf{X}_\alpha \equiv \beta \mathbf{Y}_\alpha, \alpha = 1,2,\cdots$ and same p.d. dispersion matrix $\Sigma$. Here $\beta: p \times q$ and $\Sigma$ are unknown parameters and $\mathbf{Y}_\alpha$'s are known $q \times 1$ vectors. Writing $\beta = (\beta'_1 \beta'_2)' = (\beta_{(1)}\beta_{(2)})$ with $\beta_i: p_i \times q(p_1 + p_2 = p)$ and $\beta_{(i)}: p \times q_i(q_1 + q_2 = q)$, we have constructed invariant confidence sequences for (i) $\beta$, (ii) $\beta_{(1)}$, (iii) $\beta_1$ when $\beta_2 = 0$ and (iv) $\sigma^2 = |\Sigma|$. This uses the basic ideas of Robbins (1970) and generalizes some of his and Lai's (1976) results. In the process alternative simpler solutions of some of Khan's results (1978) are obtained.