Density testing in a contaminated sample

We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z1, Z2,... which are observed under additional noise with density ψ. The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Zi. In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g = f * ψ instead on the initial density of interest f.

[1]  Tilmann Gneiting,et al.  Stochastic Models That Separate Fractal Dimension and the Hurst Effect , 2001, SIAM Rev..

[2]  Peter J. Diggle,et al.  A Fourier Approach to Nonparametric Deconvolution of a Density Estimate , 1993 .

[3]  Joseph P. Romano,et al.  Multivariate Density Estimation with General Flat-Top Kernels of Infinite Order , 1999 .

[4]  Holger Dette,et al.  Validation of linear regression models , 1998 .

[5]  W. R. van Zwet,et al.  Asymptotic Efficiency of Inverse Estimators , 2000 .

[6]  M. Rosenblatt.  A Quadratic Measure of Deviation of Two-Dimensional Density Estimates and A Test of Independence , 1975 .

[7]  Y. Nikitin,et al.  Asymptotic Efficiency of Nonparametric Tests , 1995 .

[8]  E. Mammen,et al.  Comparing Nonparametric Versus Parametric Regression Fits , 1993 .

[9]  Holger Dette,et al.  A note on the Bickel-Rosenblatt test in autoregressive time series , 2005 .

[10]  R. Carroll,et al.  Deconvolving kernel density estimators , 1987 .

[11]  Holger Dette,et al.  A consistent test for the functional form of a regression based on a difference of variance estimators , 1999 .

[12]  A. van Es,et al.  Asymptotic Normality of Nonparametric Kernel Type Deconvolution Density Estimators: crossing the Cauchy boundary , 2002, math/0212007.

[13]  P. Hall.  Central limit theorem for integrated square error of multivariate nonparametric density estimators , 1984 .

[14]  Tilmann Gneiting,et al.  Normal scale mixtures and dual probability densities , 1997 .

[15]  Efstathios Paparoditis,et al.  Spectral Density Based Goodness-of-Fit Tests for Time Series Models , 2000 .

[16]  A. J. V. Es,et al.  Simple kernel estimators for certain nonparametric deconvolution problems , 1998 .

[17]  Efstathios Paparoditis,et al.  On bootstrapping L2-type statistics in density testing , 2000 .

[18]  Estimation of integrated squared density derivatives from a contaminated sample , 2002 .

[19]  Yanqin Fan,et al.  On goodness-of-fit tests for weakly dependent processes using kernel method , 1999 .

[20]  Cun-Hui Zhang.  Fourier Methods for Estimating Mixing Densities and Distributions , 1990 .

[21]  Irène Gijbels,et al.  Practical bandwidth selection in deconvolution kernel density estimation , 2004, Comput. Stat. Data Anal..

[22]  I. A. Ibragimov,et al.  Estimation of Distribution Density Belonging to a Class of Entire Functions , 1983 .

[23]  B. K. Ghosh,et al.  The Power and Optimal Kernel of the Bickel-Rosenblatt Test for Goodness of Fit , 1991 .

[24]  Jianqing Fan.  On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems , 1991 .

[25]  P. Groeneboom,et al.  Density estimation in the uniform deconvolution model , 2003 .

[26]  P. Bickel,et al.  On Some Global Measures of the Deviations of Density Function Estimates , 1973 .

[27]  Holger Dette,et al.  Testing heteroscedasticity in nonparametric regression , 1998 .