Likelihood ratio test for the spacing between two adjacent location parameters

Suppose Yi is an estimator or sufficient statistic for a location parameter [theta]i (i = 1,...,k) of a location family. Let [theta](1) [less-than-or-equals, slant] 0(2) [less-than-or-equals, slant] ··· [less-than-or-equals, slant] [theta](k) be the ordered parameters. This paper deals with the likelihood ratio test for the hypothesis H0: [theta](k-t+1) - [theta](k-t) [less-than-or-equals, slant] [Delta] versus H1: [theta](k-t+1) - [theta](k-t) > [Delta], where [theta](k-t+1) - [theta](k-t) is the spacing between two adjacent location parameters. It is shown that if the density function of the location family has monotone likelihood ratio, the likelihood ratio test statistic is the corresponding sample spacing. However, this result may not be true for other spacings like the range [theta](k) - [theta](1).