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Principal component decomposition of non-parametric tests. (English) Zbl 0814.62026

Summary: Let \(\varphi\) denote an arbitrary nonparametric unbiased test for a Gaussian shift given by an infinite-dimensional parameter space. Then it is shown that the curvature of its power function has a principal components decomposition based on a Hilbert-Schmidt operator. Thus every test has reasonable curvature only for a finite number of orthogonal directions of alternatives.
As application, two-sided Kolmogorov-Smirnov goodness-of-fit tests are treated. We obtain lower bounds for their local asymptotic relative efficiency. They converge to one as \(\alpha \downarrow 0\) for the direction \(h_ 0 (u)= \text{sign} (2u-1)\) of the gradient of the median test. These results are analogous to earlier results of J. Hájek and Z. Šidák [Theory of rank tests. (1967; Zbl 0161.381)] for one-sided Kolmogorov-Smirnov tests.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
46N30 Applications of functional analysis in probability theory and statistics
62H25 Factor analysis and principal components; correspondence analysis

Citations:

Zbl 0161.381
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References:

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