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The power of the circular cone test: A noncentral chi‐bar‐squared distribution
Author(s) -
Conaway Mark,
Pillers Carolyn,
Robertson Tim,
Sconing James
Publication year - 1990
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315418
Subject(s) - mathematics , null distribution , test statistic , likelihood ratio test , pearson's chi squared test , statistics , chi square test , noncentral chi squared distribution , z test
Pincus (1975) derived the null distribution of the likelihood‐ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood‐ratio test statistic has a chi‐bar‐squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood‐ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi‐bar‐squared” distribution. To our knowledge, this is the first order‐restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear‐model setting. That is, the distribution of the likelihood‐ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.