Tests of fit using spacings statistics with estimated parameters

Let X1,..., Xn be a sequence of independent and identically distributed random variables with an unknown underlying continuous cumulative distribution function F. Relative to this unknown distribution function suppose one would like to test a null hypothesis concerning the goodness of fit of F to some distribution function using symmetric functions of sample spacings. In some applications the null hypothesis is simple while in others it may be composite. In this article we present the large sample theory of tests based on symmetric functions of sample spacings under composite null hypotheses and contiguous alternatives. It is shown that these test statistics have the same asymptotic distribution in the case when parameters must be estimated from the sample as in the case when parameters are specified. Optimal goodness of fit tests are also constructed for these hypotheses.