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Über die Anzahl von Zufallspunkten mit typ-gleichem nächsten Nachbarn und einen multivariaten Zwei-Stichproben-Test. (German) Zbl 0551.62027

For independent s-variate samples \(X_ 1,...,X_ m\) i.i.d. f(.), \(Y_ 1,...,Y_ n\) i.i.d. g(.), where the densities f(.), g(.) are assumed to be continuous on their respective sets of positivity, consider the number \(T_{m,n}\) of points Z of the pooled sample (which are either of ’type X’ or of ’type Y’) such that the nearest neighbor of Z is of the same type as Z. We show that, as m,n\(\to \infty\), \(m/(m+n)\to \tau\), \(0<\tau <1\), f(.)\(\equiv g(.)\), \(\lim Var T_{m,n}/\sqrt{m+n}=\sigma^ 2(\tau,s),\) independently of f(.). An omnibus test for the two sample problem ’f(.)\(\equiv g(.)\) or f(.)\(\not\equiv g(.)?'\) may be obtained by rejecting the hypothesis f(.)\(\equiv g(.)\) for large values of \(T_{m,n}\).

MSC:

62G10 Nonparametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
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References:

[1] Anderson, T.W.: Some nonparametric multivariate procedures based on statistically equivalent blocks. In: Multivariate Analysis. Hrsg. v. P.R. Krishnaiah. New York 1966, 5–27. · Zbl 0245.62054
[2] Bickel, P.J.: A distribution free version of the Smirnov two sample test in thep-variate case. Ann. Math. Statist.40, 1969, 1–23. · Zbl 0179.48704 · doi:10.1214/aoms/1177697800
[3] Bickel, P.J., undL. Breiman: Sums of functions of nearest neighbor distances, moment bounds limit theorems and a goodness of fit test. Ann. Prob.11, 1983, 185–214. · Zbl 0502.62045 · doi:10.1214/aop/1176993668
[4] Darling, D.A.: The Kolmogorov-Smirnov, Cramér-von Mises tests. Ann. Math. Statist.28, 1957, 823–838. · Zbl 0082.13602 · doi:10.1214/aoms/1177706788
[5] Friedman, J.H., undL.C. Rafsky: Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. Ann. Statist.7, 1979, 697–717. · Zbl 0423.62034 · doi:10.1214/aos/1176344722
[6] Lehmann, E.L.: Consistency and unbiasedness of certain nonparametric tests. Ann. Math. Statist.22, 1951, 165–179. · Zbl 0045.40903 · doi:10.1214/aoms/1177729639
[7] Newman, C.M., Y. Rinott, undA. Tversky: Nearest neighbors and Voronoi regions in certain Point Processes. Adv. Appl. Prob. 15, 1983, 726–751. · Zbl 0527.60050 · doi:10.2307/1427321
[8] Pickard, D.K.: Isolated nearest neighbors. J. Appl. Prob.19, 1982, 444–449. · Zbl 0486.60042 · doi:10.2307/3213499
[9] Rohlf, F.J.: Single-link clustering algorithms. In: Handbook of Statistics, Vol. 2, Hrsg. v. P.R. Krishnaiah und L.N. Kanal. Amsterdam 1982, 267–284. · Zbl 0511.62074
[10] Rosenblatt, M.: Limit theorems associated with variants of the von-Mises statistic. Ann. Math. Statist.23, 1952, 617–623. · Zbl 0048.36003 · doi:10.1214/aoms/1177729341
[11] Smirnov, N.V.: On the estimation of the discrepancy between empirical curves of distribution for two independent samples. Bull. Moscow Univ.2, 1939, 3–6.
[12] Sundrum, R.M.: On Lehmann’s two-sample test. Ann. Math. Statist.25, 1954, 139–145. · Zbl 0055.37402 · doi:10.1214/aoms/1177728853
[13] Wald, A., undJ. Wolfowitz: On a test whether two samples are from the same population. Ann. Math. Statist.11, 1940, 147–162. · Zbl 0023.24802 · doi:10.1214/aoms/1177731909
[14] Weiss, L.: Two-sample tests for multivariate distributions. Ann. Math. Statist.31, 1960, 159–164. · Zbl 0092.36401 · doi:10.1214/aoms/1177705995
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