Janssen, Arnold Principal component decomposition of non-parametric tests. (English) Zbl 0814.62026 Probab. Theory Relat. Fields 101, No. 2, 193-209 (1995). 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. Cited in 1 ReviewCited in 15 Documents 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 Keywords:spectral decomposition; unbiased test; Gaussian shift; curvature; power function; principal components decomposition; Hilbert-Schmidt operator; orthogonal directions of alternatives; two-sided Kolmogorov-Smirnov goodness-of-fit tests; lower bounds; local asymptotic relative efficiency Citations:Zbl 0161.381 PDFBibTeX XMLCite \textit{A. Janssen}, Probab. Theory Relat. Fields 101, No. 2, 193--209 (1995; Zbl 0814.62026) Full Text: DOI References: [1] Anderson, T.W., Darling, D.A.: Asymptotic theory of certain ?Goodness of fit? criteria based on stochastic processes. Ann. Math. Stat.,23, 193-212 (1952) · Zbl 0048.11301 · doi:10.1214/aoms/1177729437 [2] Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener spaces. (De Gruyter Studies in Math. 14) Berlin: 1991 · Zbl 0748.60046 [3] Drees, H., Milbrodt, H.: Components of the two-sided Kolmogorov-Smirnov test in signal detection problems with Gaussian white noise. J. Stat. Planning Inference29, 325-335 (1991) · Zbl 0747.62041 · doi:10.1016/0378-3758(91)90007-2 [4] Drees, H., Milbrodt, H.: The one-sided Kolmogorov-Smirnov test in signal detection problems with Gaussian white noise. Statistica Neerlandica48, 103-116 (1994) · Zbl 0829.62053 · doi:10.1111/j.1467-9574.1994.tb01436.x [5] Durbin, J., Knott, M.: Components of Cramér-von Mises statistics I. J. Roy. Stat. Soc. Ser. B34, 290-307 (1972) · Zbl 0238.62052 [6] Goodman, V.: Quasi-differentiable functions on Banach spaces. Proc. Am. Math. Soc.30, 367-370 (1971) · Zbl 0209.14603 [7] Gross, L.: Potential theory on Hilbert space. J. Funct. Anal.1, 123-181 (1967) · Zbl 0165.16403 · doi:10.1016/0022-1236(67)90030-4 [8] Hájek, J., Sidák, Z.: Theory of rank tests. New York: Academic Press 1967 · Zbl 0161.38102 [9] Ibragimov, I.A., Khas’minskii, R.Z.: Statistical Estimation. Asymptotic Theory. Berlin: Springer 1981 [10] Ibragimov, I. A., Khas’minskii, R. Z.: Asymptotically normal families and efficient estimation. Ann. Stat.19, 1681-1724 (1991) · Zbl 0760.62043 · doi:10.1214/aos/1176348367 [11] Inglot, T., Jurlewicz, T., Ledwina, T.: On Neyman-type smooth tests of fit. Stat.21, 549-568 (1990) · Zbl 0733.62051 · doi:10.1080/02331889008802265 [12] Janssen, A., Milbrodt, H.: Rényi type goodness of fit tests with adjusted principal direction of alternatives. Scand. J. Stat.20, 177-194 (1993) · Zbl 0798.62060 [13] Koshevnik, Yu. A., Levit, B. Ya.: On a nonparametric analogue of the information matrix. Theory Probab. Appl21, 738-753 (1976) · Zbl 0388.62037 · doi:10.1137/1121087 [14] Le Cam, L.: On a theorem of J. Hájek, Contributions to statistics. In: J. Jurecková (ed.) Hájek Memorial Volume. pp. 119-135. Dordrecht: R. Reidel 1979 [15] Le Cam, L.: Asymptotic methods in statistical decision theory. Springer Series in Statistics. New York: Springer 1986 · Zbl 0605.62002 [16] Milbrodt, H., Strasser, H.: On the asymptotic power of the two-sided Kolmogorov-Smirnov test. J. Stat. Planning Inference26, 1-23 (1990) · Zbl 0728.62049 · doi:10.1016/0378-3758(90)90091-8 [17] Neuhaus, G.: Asymptotic power properties of the Cramér-von Mises test under contiguous alternatives. J. Multivariate Anal.6, 95-110 (1976) · Zbl 0339.62035 · doi:10.1016/0047-259X(76)90022-1 [18] Pfanzagl, J., Wefelmeyer, W.: Contributions to a general asymptotic statistical theory. (Leet. Notes Stat. 13) New York: Springer · Zbl 0399.62026 [19] Pietsch, A.: Operator ideals. Amsterdam: North-Holland, 1980 · Zbl 0434.47030 [20] Pietsch, A.: Nuclear locally convex spaces. Ergebnisse der Mathem. und ihrer Grenzgeb.66, Berlin: Springer 1982 [21] Shorack, G.R., Wellner, J.A.: Empirical Processes with applications to statistics. New York: Wiley 1986 · Zbl 1170.62365 [22] Strasser, H.: Tangent vectors for models with independent but not identically distributed observation. Stat. Decisions7, 127-152 (1984) · Zbl 0686.62014 [23] Strasser, H.: Mathematical theory of statistics. Berlin: De Gruyter 1985 · Zbl 0594.62017 [24] Strasser, H.: Global extrapolations of efficiency. Stat. Decisions8, 11-26 (1990) · Zbl 0749.62032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.