Showing papers by "T. W. Anderson published in 1957"
TL;DR: In this article, the transition probabilities of a Markov chain of arbitrary order were obtained and their asymptotic distribution was obtained for a single observation of a long chain, and the relation between likelihood ratio criteria and contingency tables was discussed.
Abstract: Maximum likelihood estimates and their asymptotic distribution are obtained for the transition probabilities in a Markov chain of arbitrary order when there are repeated observations of the chain. Likelihood ratio tests and $\chi^2$-tests of the form used in contingency tables are obtained for testing the following hypotheses: (a) that the transition probabilities of a first order chain are constant, (b) that in case the transition probabilities are constant, they are specified numbers, and (c) that the process is a $u$th order Markov chain against the alternative it is $r$th but not $u$th order. In case $u = 0$ and $r = 1$, case (c) results in tests of the null hypothesis that observations at successive time points are statistically independent against the alternate hypothesis that observations are from a first order Markov chain. Tests of several other hypotheses are also considered. The statistical analysis in the case of a single observation of a long chain is also discussed. There is some discussion of the relation between likelihood ratio criteria and $\chi^2$-tests of the form used in contingency tables.
1,401 citations
TL;DR: In this paper, the authors give an approach to derive maximum likelihood estimates of parameters of multivariate normal distributions in cases where some observations are missing (Edgett [2] and Lord [3], [4]).
Abstract: S EVERAL authors recently have derived maximum likelihood estimates of parameters of multivariate normal distributions in cases where some observations are missing (Edgett [2] and Lord [3], [4]). The purpose of this note is to give an approach to these problems that indicates the estimates with a minimum of mathematical manipulation; this approach can easily be applied to other cases. (The technique bears some resemblance to that of Cochran and Bliss in a dierent problem [1].) The method will be indicated by treating the simplest case involving a bivariate normal distribution. Suppose x and y have a bivariate normal distribution with means P, and m,u variances ,2 and UY2 and correlation coefficient p. We shall indicate the density by n(x, y|,ux, p,u; 2 a2; p). Suppose n observations are made on the pair (x, y) and N-n observations are made on x; that is, N-n observations on y are missing. The data are
563 citations