Likelihood ratio test for the spacing between two adjacent location parameters

Suppose Y-i 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 equal to theta((2))less than or equal to...less than or equal to theta((k)) be the ordered parameters. This paper deals with the likelihood ratio test for the hypothesis H-0: theta((k-t+1))-theta((k-t))less than or equal to Delta versus H-1: theta((k-l+1))-theta((k-t))>Delta, where theta((k-l+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)).