The AstroStat Slog » most powerful test http://hea-www.harvard.edu/AstroStat/slog Weaving together Astronomy+Statistics+Computer Science+Engineering+Intrumentation, far beyond the growing borders Fri, 09 Sep 2011 17:05:33 +0000 en-US hourly 1 http://wordpress.org/?v=3.4 tests of fit for the Poisson distribution http://hea-www.harvard.edu/AstroStat/slog/2008/tests-of-fit-for-the-poisson-distribution/ http://hea-www.harvard.edu/AstroStat/slog/2008/tests-of-fit-for-the-poisson-distribution/#comments Tue, 29 Apr 2008 06:24:09 +0000 hlee http://hea-www.harvard.edu/AstroStat/slog/?p=280 Abstract: goodness-of-fit tests based on the Cramer-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A2, give good overall tests of fit. The statistics A2 will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.]]> Scheming arXiv:astro-ph abstracts almost an year never offered me an occasion that the fit of the Poisson distribution is tested in different ways, instead it is taken for granted by plugging data and (source) model into a (modified) χ2 function. If any doubts on the Poisson distribution occur, the following paper might be useful:

J.J.Spinelli and M.A.Stephens (1997)
Cramer-von Mises tests of fit for the Poisson distribution
Canadian J. Stat. Vol. 25(2), pp. 257-267
Abstract: goodness-of-fit tests based on the Cramer-von Mises statistics are given for the Poisson distribution. Power comparisons show that these statistics, particularly A2, give good overall tests of fit. The statistics A2 will be particularly useful for detecting distributions where the variance is close to the mean, but which are not Poisson.

In addition to Cramer-von Mises statistics (A2 and W2), the dispersion test D (so called a χ2 statistic for testing the goodness of fit in astronomy and this D statistics is considered as a two sided test approximately distributed as a χ2n-1 variable), the Neyman-Barton k-component smooth test Sk, P and T (statistics based on the probability generating function), and the Pearson X2 statistics (the number of cells K is chosen to avoid small expected values and the statistics is compared to a χ2K-1 variable, I think astronomers call it modified χ2 test) are introduced and compared to compute the powers of these tests. The strategy to provide the powers of the Cramer-von Mises statistics is that there is a parameter γ in the negative binomial distribution, which is zero under the null hypothesis (Poission distribution), and letting this γ=δ/sqrt(n) in which the parameter value δ is chosen so that for a two-sided 0.05 level test, the best test has a power of 0.5[1]. Based on this simulation study, the statistic A2 was empirically as powerful as the best test compared to other Cramer-von Mises tests.

Under the Poission distribution null hypothesis, the alternatives are overdispersed, underdispersed, and equally dispersed distributions. For the equally dispersed alternative, the Cramer-von Mises statistics have the best power compared other statistics. Overall, the Cramer-von Mises statistics have good power against all classes of alternative distributions and the Pearson X2 statistic performed very poorly for the overdispersed alternative.

Instead of binning for the modified χ2 tests[2], we could adopt A2 of W2 for the goodness-of-fit tests. Probably, it’s already implemented in softwares but not been recognized.

  1. The locally most powerful unbiased test is the statistics D (Potthoff and Whittinghill, 1966)
  2. authors’ examples indicate high significant levels compared to other tests; in other words, χ2 statistics – the dispersion test statistic D and the Pearson X2 – are insensitive to provide the evidence of the source model is not a good-fit to produce Poisson photon count data
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