New Goodness of Fit Tests Based on Stochastic EDF

Abstract

A new approach of randomization is proposed to construct goodness of fit tests generally. Some new test statistics are derived, which are based on the stochastic empirical distribution function (EDF). Note that the stochastic EDF for a set of given sample observations is a randomized distribution function. By substituting the stochastic EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling, Berk–Jones, and Einmahl–Mckeague statistics, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. In comparison to existing tests, it is shown, by a simulation study, that the new test statistics are generally more powerful than the corresponding ones based on the classical EDF or modified EDF in most cases.

Keywords:

• Anderson–Darling test
• Berk–Jones test
• Cramér–von Mises test
• Einmahl–Mckeague test
• Goodness of fit
• Kolmogorov–Smirnov test
• Stochastic empirical distribution function

Mathematics Subject Classification:

• Primary 62G10
• Secondary 62G30, 62G09

Acknowledgment

The work was supported by the National Natural Science Foundation of China (Grant No. 10771015).

Notes

^a G(1/1.2, 0.6): which is the gamma cumulative distribution function with parameters a = 1/1.2 and b = 0.6.

^a t(1): which is the Student distribution with degree of freedom 1, that is Cauchy distribution.

^b Lap(1.4): which is the Double Exponential distribution with scale parameter 1.4 and location parameter 0.

^c F _i (x) = (1 − α _i )N(0, 1) + α _i N(0, 2), i = 1,…, 5, where α₁ = 0.1, α₂ = 0.3, α₃ = 0.5, α₇ = 0.7, α₉ = 0.9.

^a F(x): the Mixture distribution 0.7N(0, 1) +0.3N(0, 2²).

^a F ₁: Weibull(1, 0.6), which is the Weibull distribution with scale parameter 1 and shape parameter 0.6.

^b F ₂: Weibull(0.7, 2), which is the Weibull distribution with scale parameter 0.7 and shape parameter 2.

^c F ₃: X = e ^Y , where Y = N(−1, 1.5²), F ₄: X = e ^Y , where Y = N(−0.5, 0.5²).

^d F ₅: F(1, 8), which is the F distribution with numerator degree of freedom 1 and denominator degrees of freedom 8.

^e G(1/3, 3): which is the gamma cumulative distribution function with parameters a = 1/3 and b = 3.