Sequential $\chi^2$- and $T^2$-Tests

Abstract

Consider a multivariate normal population with mean $\mathbf{u} = (\mu_1, \cdots, \mu_p)$ and covariance matrix $\Sigma$. Let $\mathbf{u}_0$ be a vector of constants, $_n\mathbf{\bar x}$ a vector of sample means based on $n$ observations, and $\mathbf{S}_n$ the corresponding sample covariance matrix. The statistics considered are \begin{equation*}\tag{1.1}\chi^2_n = n(_n\mathbf{\bar x} - \mathbf{u}_0)\Sigma^{-1}(_n\mathbf{\bar x} - \mathbf{u}_0)'\end{equation*} and \begin{equation*}\tag{1.2}T^2_n = n(_n\bar\mathbf{x} - \mathbf{u}_0)S^{-1}_n(_n\bar\mathbf{x} - \mathbf{u}_0)'.\end{equation*} It is shown that probability-ratio tests for a sequential test of the composite hypothesis, \begin{equation*}\tag{1.3}H_0: (\mathbf{u} - \mathbf{u}_0)\Sigma^{-1}(\mathbf{u} - \mathbf{u}_0)' = \lambda^2_0\end{equation*} against the alternative \begin{equation*}\tag{1.4}H_1: (\mathbf{u} - \mathbf{u}_0)\Sigma^{-1}(\mathbf{u} -\mathbf{u}_0)' = \lambda^2_1\end{equation*} may be based on \begin{equation*}\tag{1.5}p_{1n}/p_{0n} = \lbrack\exp - n(\lambda^2_1 - \lambda^2_0)/2\rbrack _0F_1(p/2; n\lambda^2_1\chi^2_n/4)/_0F_1(p/2; n\lambda^2_0\chi^2_n/4)\end{equation*} when $\Sigma$ is known and \begin{equation*}\tag{1.6}p_{1n}/p_{0n} = \lbrack\exp - n(\lambda^2_1 - \lambda^2_0)/2\rbrack _1F_1\lbrack n/2, p/2; n\lambda^2_1T^2_n/2(n - 1 + T^2_n)\rbrack/ \end{equation*} $_1F_1\lbrack n/2, p/2; n\lambda^2_0T^2_n/2(n - 1 + T^2_n)\rbrack$ when $\Sigma$ is unknown and must be estimated from the sample. The sequential $\chi^2$-test is associated with (1.5) and the sequential $T^2$-test with (1.6). $_0F_1$ and $_1F_1$ are respectively forms of the generalized hypergeometric function $_pF_q(a_1, \cdots, a_p; c_1, \cdots, c_q; x)$, the second being the confluent hypergeometric function. It is shown that the use of these probability ratios in sequential tests results in Type I and Type II errors of approximately $\alpha$ and $\beta$ when these values are used to obtain bounds on the probability ratios in the traditional way. It is also shown that the sequential tests terminate with probability unity. Bounds on the probability ratios are translated into bounds on $\chi^2_n$ and $T^2_n$ themselves and tables have been prepared with more tables in preparation. Procedures are also given to test sequentially whether or not two samples come from populations with the same means. The $\chi^2$-test is generalized to give simultaneous sequential tests on both the means and the covariance matrix. The average sample number functions (ASN functions) are considered and approximations to them suggested. The operating characteristic functions (OC functions) are difficult to investigate and essentially are only known approximately at $\lambda^2_0$ and $\lambda^2_1$.