## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 12, Number 1 (1984), 301-310.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346408

Digital Object Identifier
doi:10.1214/aos/1176346408

Mathematical Reviews number (MathSciNet)
MR733515

Zentralblatt MATH identifier
0565.62018

JSTOR