Hypothesis Testing Using an L 1 -Distance Bootstrap

Abstract Let X 1, X 2, …, Xn be independent random variables with common distribution function F, and let Y 1, Y 2, …, Ym be independent random variables with common distribution function G. Assume F and G are continuous and unknown. Consider the problem of testing H 0: F = G against H 1: F ≠ G. We propose a test based on the L 1 distance between density estimates for F and G. A simulation experiment is used to compare this statistic to others under both bootstrap and permutation resampling procedures. The well-known Kolmogorov–Smirnov test is also used for comparison. The L 1-distance method compares quite favorably.

[1]  F. Massey,et al.  The Distribution of the Maximum Deviation Between two Sample Cumulative Step Functions , 1951 .

[2]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[3]  D. Darling.  The Kolmogorov-Smirnov, Cramer-von Mises Tests , 1957 .

[4]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[5]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[6]  J. H. Schuenemeyer,et al.  A Modified Kolmogorov-Smirnov Test Sensitive to Tail Alternatives , 1983 .

[7]  L. Devroye,et al.  Nonparametric Density Estimation: The L 1 View. , 1985 .

[8]  A. Bowman.  A comparative study of some kernel-based nonparametric density estimators , 1985 .

[9]  B. Silverman.  Density estimation for statistics and data analysis , 1986 .

[10]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[11]  Matt P. Wand,et al.  Minimizing L 1 distance in nonparametric density estimation , 1988 .

[12]  D. W. Scott,et al.  Nonparametric Estimation of Probability Densities and Regression Curves , 1988 .

[13]  Scott M. Smith,et al.  Computer Intensive Methods for Testing Hypotheses: An Introduction , 1989 .

[14]  Central Limit Theorems for $L_p$ Distances of Kernel Estimators of Densities Under Random Censorship , 1991 .

[15]  Adrian Bowman,et al.  Density based tests for goodness-of-fit , 1992 .

[16]  A. Bowman,et al.  Adaptive Smoothing and Density-Based Tests of Multivariate Normality , 1993 .

[17]  R. Tibshirani,et al.  An introduction to the bootstrap , 1993 .

[18]  N. H. Anderson,et al.  Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates , 1994 .

[19]  W. D. Kaigh.  Distribution-Free Two-Sample Tests Based on Rank Spacings , 1994 .