The Annals of Statistics

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

B. K. Sinha and S. K. Sarkar

Full-text: Open access

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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 62F25: Tolerance and confidence regions
Secondary: 62L10: Sequential analysis 62H99: None of the above, but in this section

Keywords
Confidence sequences likelihood ratio martingales multivariate normal distribution maximal invariant

Citation

Sinha, B. K.; Sarkar, S. K. Invariant Confidence Sequences for Some Parameters in a Multivariate Linear Regression Model. Ann. Statist. 12 (1984), no. 1, 301--310. doi:10.1214/aos/1176346408. https://projecteuclid.org/euclid.aos/1176346408


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