The Use of Posterior Predictive P-Values in Testing Goodness-of-Fit

In this article, we introduce two goodness-of-fit tests for testing normality through the concept of the posterior predictive p-value. The discrepancy variables selected are the Kolmogorov-Smirnov (KS) and Berk-Jones (BJ) statistics and the prior chosen is Jeffreys’ prior. The constructed posterior predictive p-values are shown to be distributed independently of the unknown parameters under the null hypothesis, thus they can be taken as the test statistics. It emerges from the simulation that the new tests are more powerful than the corresponding classical tests against most of the alternatives concerned.

[1]  AN Kolmogorov-Smirnov,et al.  Sulla determinazione empírica di uma legge di distribuzione , 1933 .

[2]  F. Massey.  The Kolmogorov-Smirnov Test for Goodness of Fit , 1951 .

[3]  I. Guttman.  The Use of the Concept of a Future Observation in Goodness‐Of‐Fit Problems , 1967 .

[4]  Douglas H. Jones,et al.  Goodness-of-fit test statistics that dominate the Kolmogorov statistics , 1979 .

[5]  D. Rubin.  Bayesianly Justifiable and Relevant Frequency Calculations for the Applied Statistician , 1984 .

[6]  Ralph B. D'Agostino,et al.  Goodness-of-Fit-Techniques , 2020 .

[7]  K. Koehler,et al.  Goodness-of-fit tests based on P—P probability plots , 1990 .

[8]  K. Koehler,et al.  Goodness-of-Fit Tests Based onP-PProbability Plots , 1990 .

[9]  Xiao-Li Meng,et al.  Posterior Predictive $p$-Values , 1994 .

[10]  Xiao-Li Meng,et al.  Posterior Predictive Assessment of Model Fitnessvia Realized , 1995 .

[11]  Xiao-Li Meng,et al.  POSTERIOR PREDICTIVE ASSESSMENT OF MODEL FITNESS VIA REALIZED DISCREPANCIES , 1996 .

[12]  M. T. Rodríguez-Bernal,et al.  Asymptotic behaviour of the posterior predictive p-value , 1997 .

[13]  J. Horra,et al.  The posterior predictive p-value: an alternative to the classical p-value , 1999 .

[14]  M. J. Bayarri,et al.  P Values for Composite Null Models , 2000 .

[15]  M. T. Rodríguez-Bernal,et al.  Posterior predictive p-values: what they are and what they are not , 2001 .

[16]  H. Stern,et al.  Posterior predictive model checking in hierarchical models , 2003 .

[17]  L. Leemis,et al.  Minimum Kolmogorov–Smirnov test statistic parameter estimates , 2006 .

[18]  N. Hjort,et al.  Post-Processing Posterior Predictive p Values , 2006 .

[19]  Zvi Drezner,et al.  A Modified Kolmogorov–Smirnov Test for Normality , 2010, Commun. Stat. Simul. Comput..

[20]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .