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

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


62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
Full Text: EuDML


[1] R. J. Beran: Linear rank statistics under alternatives indexed by a vector parameter. Ann. Math. Statist. 41 (1970), 1896-1905. · Zbl 0231.62064
[2] D. R. Cox D. V. Hinkley: Theoretical Statistics. London, Chapman and Hall, 1974. · Zbl 0334.62003
[3] B. S. Duran W. W. Tsai T. S. Lewis: A class of location-scale nonparametric tests. Biometrika 63 (1976), 113-176. · Zbl 0333.62029
[4] M. N. Goria: A survey of two sample location-scale problem: asymptotic relative efficiencies of some rank tests. Statistica Neerlandica 36 (1982), 3-13. · Zbl 0488.62028
[5] J. Hájek Z. Šidák: Theory of Rank Tests. New York, Academic Press, 1967. · Zbl 0161.38102
[6] Y. Lepage: A combination of Wilcoxon and Ansari-Bradley statistics. Biometrika 58 (1971), 213-217. · Zbl 0218.62039
[7] Y. Lepage: Asymptotically optimum rank tests for contiguous location-scale alternative. Commun. Statist. Theor. Meth. A 4 (7) (1975), 671-687. · Zbl 0308.62033
[8] Y. Lepage: Asymptotic power efficiency for a location-scale problem. Commun. Statist. Theor. Meth., A 5 (13) (1976), 1257-1274. · Zbl 0351.62030
[9] Y. Lepage: A class of nonparametric tests for location-scale parameter. Commun. Statist. Theor. Meth. A 6 (7) (1977), 649-659. · Zbl 0381.62035
[10] R. H. Randles R. V. Hogg: Certain uncorrelated and independent rank statistics. JASA 66 (1971), 569-574. · Zbl 0243.62032
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