On the most powerful location and scale invariant tests based on an incomplete set of order statistics (Q1269932)

The problem of goodness-of-fit testing in the case of two-parameter distribution families often occurs in practice. We shall suppose that unknown parameters are the location and scale. In this case there exist most powerful tests that are invariant under translation and change of scale. Let \(Y_1, \dots, Y_r\) be a sample with joint density \(p\) (null hypothesis) and \(q\) (alternative). We test this hypothesis against this alternative. The critical region of the most powerful invariant test is \[ \begin{aligned} \widehat q(y_1, \dots, y_r)\equiv & \int_0^{+\infty} \lambda^{r-2} d\lambda \int^{+\infty}_{- \infty} q(\lambda y_1- u,\dots, \lambda y_r-u) du\\ >C_\alpha & \int_0^{-\infty} \lambda^{r-2} d\lambda \int^{+\infty}_{-\infty} p(\lambda y_1- u,\dots, \lambda y_r-u)du \equiv \widehat p(y_1, \dots, y_r), \end{aligned} \] where the constant \(C_\alpha\) depends on the given significance level \(\alpha(0 <\alpha<1)\). Note that for this test a complete sample is needed. However, there are many practical situations in which only an incomplete set of order statistics (s.o.s.) is observable. In this paper, the critical regions of the most powerful location and scale invariant tests based on incomplete s.o.s. are presented in the case of two-parameter distributions. We consider only uniform, exponential, negative exponential, and Laplace (double exponential) distributions.