Goodness of fit tests via exponential series density estimation

The properties of a new nonparametric goodness of fit test are explored. It is based on a likelihood ratio test, applied via a consistent series density estimator in the exponential family. The focus is on its computational and numerical properties. Specifically it is found that the choice of approximating basis is not crucial and that the choice of model dimension, through data-driven selection criteria, yields a feasible, parsimonious procedure. Numerical experiments show that the new tests have significantly more power than established tests, whether based upon the empirical distribution function, or alternate density estimators.

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

[2]  B. R. Crain.  More on Estimation of Distributions Using Orthogonal Expansions , 1976 .

[3]  Nils Lid Hjort,et al.  Goodness of Fit via Non‐parametric Likelihood Ratios , 2004 .

[4]  B. R. Crain,et al.  Estimation of Distributions Using Orthogonal Expansions , 1974 .

[5]  A. Barron,et al.  APPROXIMATION OF DENSITY FUNCTIONS BY SEQUENCES OF EXPONENTIAL FAMILIES , 1991 .

[6]  M. Stephens.  EDF Statistics for Goodness of Fit and Some Comparisons , 1974 .

[7]  Binning methodology for nonparametric goodness-of-fit test , 2003 .

[8]  B. R. Crain.  An Information Theoretic Approach to Approximating a Probability Distribution , 1977 .

[9]  H. Akaike.  A new look at the statistical model identification , 1974 .

[10]  D. Lawley.  A GENERAL METHOD FOR APPROXIMATING TO THE DISTRIBUTION OF LIKELIHOOD RATIO CRITERIA , 1956 .

[11]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[12]  S. Portnoy.  Asymptotic Behavior of Likelihood Methods for Exponential Families when the Number of Parameters Tends to Infinity , 1988 .

[13]  W. J. Conover,et al.  Practical Nonparametric Statistics , 1972 .

[14]  S. Shapiro,et al.  A Comparative Study of Various Tests for Normality , 1968 .

[15]  D. Haughton.  On the Choice of a Model to Fit Data from an Exponential Family , 1988 .

[16]  S. Shapiro,et al.  An Analysis of Variance Test for Normality (Complete Samples) , 1965 .

[17]  T. W. Anderson,et al.  Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes , 1952 .