A Cramer-von Mises type goodness-of-fit test with asymmetric weight function. The Gaussian and exponential cases

A goodness of fit test of the Cramer - von Mises type which gives more weight to the upper (lower) tail of the distribution was proposed and studied by the authors, Rodriguez and Viollaz (1995), for the case of a simple null hypothesis. In the present paper that: goodness of fit test is studied for a composite null hypothesis in the case of a Gaussian distribution with one or two unknown parameters and the exponential distribution with unknown parameter. Asymptotic percentage points of the distribution function of the test statistic are obtained for the cases studied: Case 1: Normal distribution with known variance, Case 2: Normal distribution with known mean, Case 3: normal distribution with unknown mean and variance and Case 4: exponential distribution with unknown parameter. For all cases asymptotic means and variances of the goodness of fit statistic are given, as well as some weights of the decomposition of the test as a sum of weighted chi(2)(1) variables.