ON the Robustness of Mixture Index of Fit

The aim of this paper is to investigate the mixture index of fit in hypothesis testing problems from the point of view of robustness. The concept of contamination plot is introduced and an algorithm is proposed to determine it. Our algorithm is a remarkable application of the widely used EM algorithm by involving a two-phase M-step procedure. In the parametric phase the parameters of the model in the null hypothesis are estimated using the maximum likelihood method, while in the nonparametric phase the contaminating distribution is determined by a filling technique. It is proved that the objective function decreases monotonically during the iterations. Finally, the algorithm is applied and discussed when the hypothesis of independence is tested for contingency tables.

[1]  C. T. Rajagopal.  Some limit theorems , 1948 .

[2]  R. R. Bahadur.  Stochastic comparison of tests , 1960 .

[3]  P. J. Huber,et al.  Minimax Tests and the Neyman-Pearson Lemma for Capacities , 1973 .

[4]  R. Snee.  Graphical Display of Two-way Contingency Tables , 1974 .

[5]  G. Tusnády.  On Asymptotically Optimal Tests , 1977 .

[6]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[7]  Paul H. Zipkin,et al.  Simple Ranking Methods for Allocation of One Resource , 1980 .

[8]  Peter J. Huber,et al.  Robust Statistics , 2005, Wiley Series in Probability and Statistics.

[9]  P. Diaconis,et al.  Testing for independence in a two-way table , 1985 .

[10]  R. R. Bahadur.  Some Limit Theorems in Statistics , 1987 .

[11]  D. Donoho,et al.  The "Automatic" Robustness of Minimum Distance Functionals , 1988 .

[12]  Toshihide Ibaraki,et al.  Resource allocation problems - algorithmic approaches , 1988, MIT Press series in the foundations of computing.

[13]  Timothy R. C. Read,et al.  Goodness-Of-Fit Statistics for Discrete Multivariate Data , 1988 .

[14]  Sanford Weisberg,et al.  Directions in Robust Statistics and Diagnostics , 1991 .

[15]  Marianthi Markatou,et al.  Robust M-type testing procedures for linear models , 1991 .

[16]  B. Lindsay,et al.  A New Index of Fit Based on Mixture Methods for the Analysis of Contingency Tables , 1994 .

[17]  E. Ronchetti,et al.  Robust Bounded-Influence Tests in General Parametric Models , 1994 .

[18]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[19]  B. Lindsay,et al.  A Note on Calculating the π* Index of Fit for the Analysis of Contingency Tables , 1996 .

[20]  Naoki Katoh,et al.  Resource Allocation Problems , 1998 .

[21]  FINDING AND CHARACTERIZATION OF L CAL PTIMA IN THE đ .PROBLEM FOR TWO-WAY CONTINGENCY TABLES , 2000 .

[22]  I. Csiszár,et al.  Divergence minimization under prior inequality constraints , 2001, Proceedings. 2001 IEEE International Symposium on Information Theory (IEEE Cat. No.01CH37252).

[23]  A. Formann.  Testing the Rasch model by means of the mixture fit index. , 2006, The British journal of mathematical and statistical psychology.

[24]  J. Revuelta.  Estimating the pi* goodness of fit index for finite mixtures of item response models. , 2008, The British journal of mathematical and statistical psychology.

[25]  B. Lindsay,et al.  Building and using semiparametric tolerance regions for parametric multinomial models , 2009, 0909.0608.