The power of the circular cone test: A noncentral chi‐bar‐squared distribution

Abstract 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.