scispace - formally typeset
Search or ask a question

Showing papers by "T. W. Anderson published in 1964"



Journal ArticleDOI
TL;DR: In this paper, sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained, and the likelihood-ratio test, Lawley-Hotelling trace test, and Roy's maximum root test satisfy these conditions.
Abstract: The test procedures, invariant under certain groups of transformations [4], for testing a set of multivariate linear hypotheses in the linear normal model depend on the characteristic roots of a random matrix. The power function of such a test depends on the characteristic roots of a corresponding population matrix as parameters; these roots may be regarded as measures of deviation from the hypothesis tested. In this paper sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1], Lawley-Hotelling trace test [1], and Roy's maximum root test [6] satisfy these conditions. The monotonicity of the power function of Roy's test has been shown by Roy and Mikhail [5] using a geometrical method.

62 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived a sufficient condition for a test depending on the roots of a sample covariance matrix to have the power function monotonically increasing in each root of the matrix, where the acceptance region of the test has the property that any point with coordinates not greater than these coordinates is also in the region.
Abstract: Invariant tests of the hypothesis that $\mathbf\Sigma_1 = \Sigma_2$ are based on the characteristic roots of $S_1S^{-1}_2$, say $c_1 \geqq c_2 \geqq \cdots \geqq c_p$, where $\Sigma_1$ and $\Sigma_2$ and $\mathbf{S}_1$ and $\mathbf{S}_2$ are the population and sample covariance matrices, respectively, of two multivariate normal populations; the power of such a test depends on the characteristic roots of $\Sigma_1\Sigma^{-1}_2$. It is shown that the power function is an increasing function of each ordered root of $\Sigma_1\Sigma^{-1}_2$ if the acceptance region of the test has the property that if $(c_1, \cdots, c_p)$ is in the region then any point with coordinates not greater than these, respectively, is also in the region. Examples of such acceptance regions are given. For testing the hypothesis that $\Sigma = I$, a similar sufficient condition is derived for a test depending on the roots of a sample covariance matrix $\mathbf{S}$, based on observations from a normal distribution with covariance matrix $\Sigma$, to have the power function monotonically increasing in each root of $\Sigma$.

40 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered a special case of the situation in which after the decision to stop observation is made a fixed number of additional observations are available for the terminal decision, and the test is then based on the ratio of the likelihoods (at two simple hypotheses) of all the observations (whatever stopping rule is used).
Abstract: In testing one hypothesis against another, observations may be obtained sequentially. The special feature of the situation considered in this paper is that after the decision to stop observation is made a fixed number of additional observations are available for the terminal decision. The test is then based on the ratio of the likelihoods (at two simple hypotheses) of all the observations (whatever stopping rule is used). Numerical comparisons are made for various cases when the hypotheses concern the mean of a normal distribution with the variance known; these cases include a fixed-sample-size procedure as well as cases where several numbers of additional observations are available after the Wald sequential probability ratio test is used as a stopping rule.

36 citations


Journal ArticleDOI
TL;DR: In this paper, sufficient conditions on the invariant procedure for the power function to be a monotonically increasing function of each of the parameters are obtained, and the likelihood-ratio test and Roy's maximum root test satisfy these conditions.
Abstract: For testing independence between two sets of normally distributed variates we consider the class of test procedures which are invariant under certain groups of transformations and depend only on the sample canonical correlation coefficients [1]. The power function of such a test depends only on the population canonical correlation coefficients as parameters, which may be regarded as measures of deviation from the hypothesis. In this paper sufficient conditions on the invariant procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1] and Roy's maximum root test [5] satisfy these conditions. In [4] only the unbiasedness of the maximum root test was proved, although the authors claimed to prove the monotonicity property.

34 citations