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

Abstract

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.

Keywords:

• Berk-Jones test
• Discrepancy variable
• Goodness-of-fit test
• Jeffreys’ piror
• Kolmogorov-Smirnov test
• Posterior predictive p-value

Mathematics Subject Classification:

• Primary 62G10
• Secondary 62F15

Acknowledgments

The authors are grateful to the two reviewers and the Editor-in-Chief for their valuable comments and suggestions. The work is supported by the National Natural Science Foundation of China (Grant Nos. 11201005, 11071015, 11126197), the Natural Science Foundation of Anhui Province (Grant No. 1308085QA13), the Project of Natural Science Foundation of of Universities in Anhui Province (Grant No. KJ2012A135) and the Key Project of Distinguished Young Talents of Universities in Anhui Province (Grant No. 2012SQRL028ZD).

Notes

^aThe Uniform distribution on the interval (0, 1).

^b B(a, b), the Beta distribution with pdf

^c Tu(λ), the Tukey distribution defined by U ^λ − (1 − U)^λ, where U is the uniform distribution on the interval (0, 1).

^d TrN(a, b), the truncated standard normal distribution with pdf , where φ(x) is the pdf of the standard normal.

^e SB(a, b) is defined by , where Z is the standard normal.

^a t(k), the Student distribution with k degrees of freedom.

^b The Logistic distribution with pdf f(x) = e ^−x (1 + e ^−x )⁻², x ∈ (− ∞, ∞).

^c SU(a, b), the Johnson distribution defined by sinh((Z − a)/b), where Z is the standard normal.

^d The Laplace distribution with pdf

^a χ²(k), the Chi-squared distribution with degree of freedom k.

^b Gam(λ), the Gamma distribution with pdf .

^c LogN(λ), the LogNoraml distribution defined by e ^λZ , where Z is the standard normal.

^d Weib(k), the Weibull distribution with pdf f(x) = kx ^k−1 e ^{−x k} , x > 0.