Scale invariant for one-sided multivariate likelihood ratio tests

Abstract

Suppose 1 2 , ,..., n X X X is a random sample from Np ( ,V ) distribution. Consider 0 1 2 : ... 0 p H      and1 : 0 for 1, 2,..., i H   i  p , let 1 0 H  H denote the hypothesis that 1 H holds but 0 H does not, and let ~ 0 H denote thehypothesis that 0 H does not hold. Because the likelihood ratio test (LRT) of 0 H versus 1 0 H  H is complicated, severalad hoc tests have been proposed. Tang, Gnecco and Geller (1989) proposed an approximate LRT, Follmann (1996) suggestedrejecting 0 H if the usual test of 0 H versus ~ 0 H rejects 0 H with significance level 2 and a weighted sum of the samplemeans is positive, and Chongcharoen, Singh and Wright (2002) modified Follmann’s test to include information about thecorrelation structure in the sum of the sample means. Chongcharoen and Wright (2007, 2006) give versions of the Tang-Gnecco-Geller tests and Follmann-type tests, respectively, with invariance properties. With LRT’s scale invariant desiredproperty, we investigate its powers by using Monte Carlo techniques and compare them with the tests which we recommendin Chongcharoen and Wright (2007, 2006).

Keywords

• Follmann’s test
• likelihood ratio tests
• modified Follmann’s test
• Tang-Gnecco-Geller test