Goodness-of-fit testing: the thresholding approach

The classical Pearson's chi-square test for goodness-of-fit has found extensive applications in areas such as contingency tables and, recently, multiple testing. Mann and Wald [(1942), ‘On the Choice of the Number of Class Intervals in the Application of the Chi Square Test’, The Annals of Mathematical Statistics, 13, 306–317] were the first to establish the power advantages of letting the number nbin of bins tend to infinity with n, and found nbin=n2/5 to be the optimal rate. For a corresponding development in the area of contingency tables, see Holst [(1972), ‘Asymptotic Normality and Efficiency for Certain Goodness-of-Fit Tests’, Biometrika, 59, 137–145], Morris [(1975), ‘Central Limit Theorems for Multinomial Sums’, The Annals of Statistics, 3, 165–188], and Koehler and Larntz [(1980), ‘An Empirical Investigation of Goodness-of-Fit Statistics for Sparse Multinomials’, Journal of the American Statistical Association, 75, 336–344]. In this paper, we consider the use of thresholding methods to further improve on the power of Pearson's chi-square test. An alternative statistic, based on the cell averages, is also studied. The Fourier or wavelet transformation is used to ensure power enhancement in both high- and low-signal-to-noise ratio alternatives. Simulations suggest that application of order thresholding (Kim, M.H., and Akritas, M.G. (2010), ‘Order Thresholding’, The Annals of Statistics, 38, 2314–2350) achieves accurate type I error rates, and competitive power.