Testing the hypothesis concerning the type of distribution and normed order statistics in a triangular array scheme (Q1269925)

Let \(X_n= \{X_j^{(n)} ,j=1, \dots,n\}\), \(n\geq 3\), be a standard sample with \({\mathbf P}\{X_j^{(n)} <x\}= F_n(x)\), and let \(t_j(X_n)\) be the \(j\)th order statistic. Below we shall investigate the well-known \((\lambda_0, \lambda_1)\)-normed order statistics \[ Y_{jn} (k_0, k_1) ={t_j(x_n) -t_{k_0} (x_n) \over t_{k_1} (x_n)- t_{k_0} (x_n)},\;k_0\leq j\leq k_1, \tag{1} \] with \(k_0 =[\lambda_0(n+1)]\), \(k_1=n+1-[(1-\lambda_1)(n+1)]\). Here \([\alpha]\) is the integer part of \(\alpha\). Note that the statistics (1) are translation and scale invariant. We shall concentrate on the ``close'' alternatives \[ F_n= \left(1-{\alpha \over \sqrt n}\right) F+ {\alpha \over \sqrt n} G. \tag{2} \] There are several reasons for this special interest. The first is to obtain a nontrivial limit of the power of the test based on statistics (1), as follows from considering alternatives (2). The second reason is connected with the Tukey-Huber contamination model, where the ``real'' distribution is \(F=(1- \varepsilon) F_0+ \varepsilon G_0\), and \(F_0\) is the ``true'' distribution, \(\varepsilon\) is the contamination part. In practice, there are many cases for which one can take \(\varepsilon =\alpha/ \sqrt n\) of \(\alpha= \varepsilon \sqrt n\).