Non-unbiased two-sample nonparametric tests. Numerical example.

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1 Non-unbiased two-sample nonparametric tests. Numerical example. Xeniya Yermolenko Charles University in Prague, Department of Statistics SUMMER SCHOOL ROBUST 2016, 1116 SEPTEMBER September 13, 2016 X. Yermolenko, Charles University in Prague p. 1 out of 14

2 Rationale Lehmann E.L Testing statistical hypotheses Sugiura et al. 1965, 2006 Biased and unbiased two-sided Wilcoxon tests Liu and Singh 1993 Proposed a two-sample test of Wilcoxon type, based on the ranks of depths of the data Amrhein, P An example of a two-sided Wilcoxon signed rank test which is not unbiased Jure ková et al. 2002, 2012 The two-sample multivariate testing problem X. Yermolenko, Charles University in Prague p. 2 out of 14

3 Outline Two-sample Wilcoxon test The power function and the signicance level. Basic denitions Distribution Numerical illustration Numerical example Simulations Conclusion Bibliography Contacts X. Yermolenko, Charles University in Prague p. 3 out of 14

4 Two-sample Wilcoxon test Let X 1,..., X n F (x) and Y 1,..., Y m G(x) be two random samples from the absolutely continuous distributions. Let us assume that G(x) = F (x ). We want to test the hypothesis against the two-sided alternative: H 0 : = 0, H 1 : 0, using the following two-sample Wilcoxon test: { 1, if X (n) < Y (1) or X (1) > Y (m), φ(x 1,..., X n, Y 1,..., Y m) = 0, otherwise. X. Yermolenko, Charles University in Prague p. 4 out of 14

5 The power function and the signicance level. Basic denitions The power function of φ is given by: β n,m( ) = P (X (n) < Y (1) ) + P (X (1) > Y (m) ) = G m (x)d(1 (1 F (x)) n ) = + F m (x )d(1 (1 F (x)) n ). The signicance level α of this test is: + F n (x)d(1 (1 G(x)) m )+ F n (X)d(1 (1 F (x )) m )+ α = P H0 (X (n) < Y (1) ) + P H0 (X (1) > Y (m) ) = β n,m(0) = 2m!n! (m + n)!. We have a property of the power function β n,m( ) = β m,n( ). Therefore, everywhere below we will assume that n m. The test φ is unbiased if it satises inf βn,m( ) α. X. Yermolenko, Charles University in Prague p. 5 out of 14

6 Distribution Suppose that F is a Skew Logistic Distribution with a scale parameter b > 0 and a skew parameter λ > 0 e f(x) = 2λ x/bλ, if x < 0, b(1 + e x/bλ ) λ 2 e xλ/b, if x 0. b(1 + e xλ/b ) 2 2λ 2 1, if x < 0, F (x) = 1 + λ e x/bλ λ λ e xλ/b, if x λ e xλ/b where f(x) is a density function of the X. X. Yermolenko, Charles University in Prague p. 6 out of 14

7 Distribution All graphics were built in scilab Figure: PDF of a skew logistic distribution for dierent values of λ, for b = 0.01 Unless otherwise stipulated, throughout b = 0.01 and λ = 2.5 X. Yermolenko, Charles University in Prague p. 7 out of 14

8 Numerical illustration All computations were made in scilab If n = m, then β n,m( ) symmetric about = 0 and the test φ is unbiased. Otherwise β n,m( ) skew and the test φ is biased. n m α βteor min min e e e 04 β min teor minimum of the numerically computed function β n,m( ); min the point at which this minimum was obtained. X. Yermolenko, Charles University in Prague p. 8 out of 14

9 Numerical illustration Figure: The power function β n,m( ) for n=m=3 and for n=19, m=1. This example is very important, because in both cases the signicance level of the test φ is equal to 0.1, meanwhile in the rst case the test is unbiased but in the second it is biased. X. Yermolenko, Charles University in Prague p. 9 out of 14

10 Numerical illustration On the following plots the power functions are shown for dierent combinations of n and m. X. Yermolenko, Charles University in Prague p. 10 out of 14

11 Simulation n m α N k 0 β prac e e e 04 0 [ min; 0] a point for simulation of sample Y 1,..., Y m N the number of simulations of samples X 1,..., X n and Y 1,..., Y m; k 0 the number of experiments in which φ(x 1,..., X n, Y 1,..., Y m) = 1 β prac = k0 N empirical power of the test φ. X. Yermolenko, Charles University in Prague p. 11 out of 14

12 Conclutions In the case of the skew logistic distribution with a scale parameter b > 0 and a skew parameter λ > 0, λ 1 we have if m = n, then the test φ is unbiased if m n, then the test φ is biased. X. Yermolenko, Charles University in Prague p. 12 out of 14

13 Bibliography 1) Lehmann E.L. (1986). Testing statistical hypotheses. Chapman & Hall ) Amrhein, P. (1995). An example of a two-sided Wilcoxon signed rank test which is not unbiased. Ann. Inst. Statist. Math ) Sugiura, N., Murakami, H., Lee, S.K. & Maeda, Y. (2006). Biased and unbiased two-sided Wilcoxon tests for equal sample sizes. Ann. Inst. Statist. Math. 58, ) Jure ková, J., & Kalina, J. (2012). Nonparametric multivariate rank tests and their unbiasedness. Bernoulli 18, ) Jure ková, J. & Milhaud, X. (2003). Derivative in the mean of a density and statistical applications. IMS Lecture Notes 42, ) D. Sastry, D. Bhati (2014) A New Skew Logistic Distribution: Properties and Applications. Brazilian Journal of Probab. and Statist. Dec X. Yermolenko, Charles University in Prague p. 13 out of 14

14 Contacts Contact information Address Charles University in Prague, Department of Statistics, Sokolovská 83, , Czech Republic Acknowledgments This work was supported by grants SVV , GACR S, PRVOUK X. Yermolenko, Charles University in Prague p. 14 out of 14