On the asymptotic properties of rank statistics for the two-sample location and scale problem. (English) Zbl 0614.62052

Goria, M. N.; Vorličková, Dana

Apl. Mat. 30, 425-434 (1985).

Let \((X_ 1,...,X_ m)\) and \((X_{m+1},...,X_ N)\), \(N=m+n\), be two independent random samples, and suppose that for some unknown \(\theta =(\theta_ 1,\theta_ 2)\in {\mathbb{R}}^ 2\), the variable \(X_ 1\) has density f(x) and \(X_{m+1}\) density \(e^{-\theta_ 2}f(e^{-\theta_ 2}(x-\theta_ 1))\). Various authors investigated quadratic forms of linear rank statistics \(S_ 1\) and \(S_ 2\), for testing the hypothesis \(H: \theta\) \(=0\). Here \(S_ 1(S_ 2)\) is a suitable rank statistic for testing the difference in location (in scale). It is well-known that, under natural conditions for the (optimal) scores, \(S_ 1\) and \(S_ 2\) are uncorrelated under H.
In the present paper it is proved that \(S_ 1\) and \(S_ 2\) are asymptotically independent under H as well as under contiguous alternatives, when, in addition, f is assumed to be symmetric about the origin. This result turns out to be very useful when the asymptotic power of tests based on quadratic forms in \(S_ 1\) and \(S_ 2\) is considered.

Reviewer: R.Helmers

MSC:

62G10	Nonparametric hypothesis testing
62E20	Asymptotic distribution theory in statistics

Keywords:

hypothesis of randomness; two-sample location-scale problem; quadratic forms of linear rank statistics; asymptotically independent; contiguous alternatives; asymptotic power

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