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Local asymptotics for linear rank statistics with estimated score functions. (English) Zbl 0632.62045

Let \(X_ 1,...,X_ m\), \(Y_ 1,...,Y_ n\) be independent real-valued random variables, the \(X_ i's\) with common distribution (df) F and the \(Y_ i's\) with df G. The null-hypothesis of randomness \(H_ 0:\) \(F=G\) is tested against the “stochastically larger” alternative \(H_ 1:\) \(F\leq G\), \(F\neq G\). Since \(F\leq G\) is equivalent to \(F(x)=G(x-D(x))\) for all x for some shift function \(D: {\mathbb{R}}\to [0,\infty)\), \(H_ 1\) can be interpreted as shift alternative where the size of the shift may vary for different parts of the distribution.
Classically D(x) is constant and we are interested in testing \(H_ 0\) against the much more restricted alternative \(G(x)=F(x-D)\) for some constant D and all x. In this special case it is well-known that the order statistics of the combined sample \(X_ 1,...,X_ m\), \(Y_ 1,...,Y_ n\) can be used to estimate the score function \[ \phi (u,f)=- f'(F^{-1}(u))/f(F^{-1}(u)),\quad 0<u<1, \] related to F, in order to obtain an asymptotically optimal rank test for testing \(H_ 0\) against classical shift alternatives.
For testing \(H_ 0\) against \(H_ 1\), however, the order statistics will contain no useful information for choosing between \(H_ 0\) and \(H_ 1\). For this more general situation the author replaces the parameter (F,G) for his testing problem by an equivalent one based on a function \(b=(mn/N)^{1/2}\bar b\), where \(\bar b=\bar f-\bar g\) with \[ \bar f=dF_ 0H^{-1}/d\lambda,\quad \bar g=dG_ 0H^{-1}/d\lambda. \] Here \(H=(m/N)F+(1-m/N)G\) and \(\lambda\) denotes Lebesgue measure on (0,1). In discussing local asymptotic results the author employs (b,H) instead of (F,G). Here H is a nuisance parameter, whereas the function b contains all information contained in (F,G) which determines whether \(H_ 0\) or \(H_ 1\) is true.
In a previous paper by K. Behnen, the author and F. Ruymgaart [ibid. 11, 1175-1189 (1983; Zbl 0548.62029)] an estimator \(\hat b{}_ N\) of b is proposed, using a kernel method based on ranks and a quadratic rank statistic \(S_ N(\hat b_ N)\) is used for testing \(H_ 0\) against the general alternative \(F\neq G\). In the present paper the behaviour of the corresponding test as well as of a variant adapted to the testing problem \((H_ 0,H_ 1)\) under consideration is investigated by means of local asymptotic results with the bandwidth of the kernel kept fixed. It turns out that this type of asymptotics fits finite sample Monte-Carlo results better than previous results do. Also the power behaviour of the tests are explained better in this way. Some recommendations for the application of these tests in practice are also given.
Reviewer: R.Helmers

MSC:

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0548.62029
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