Asymptotics for Generalized Chi-Square Goodness-of-Fit Tests.
研究了多种卡方拟合优度检验统计量的渐近行为，包括类数选择、估计方法影响和功效近似，为统计实践提供理论指导。
This monograph considers the behaviour of various types of chisquare goodness-of-fit test statistics.The first chapter gives a review of recent literature on the subject.i Chapter II investigates the influence of the number of classes kin the presence of a location-scale nuisance parameter.When k ➔ 00 , we prove 2 the asymptotic normality of the Moore-Spruill (1975) class of X statistics (extending Morris' (1975) theorem for the Pearson statistic when k ➔ 00 and no nuisance parameters are present).Criteria are developed whether to choose a large or a small number of classes.A theoretical explanation was still lacking for simulations showing that the Rao-Robson-Nikulin test dominates other commonly used x 2 tests.The present limit theorem implies that, when k ➔ 00 , the Rao-Robson-Nikulin test is better in the sense of Pitman efficiency.The choice of the location-scale estimator is the subject of the third chapter.Non-robust estimation (i.e. the estimator is not £-consistent under local alternatives) is best: under non-robust estimation general EDF tests, including generalized x 2 tests, are consistent while the asymptotic local power remains bounded away from one in more classical situations of e.g .. ML or robust estimation under heavy-tailed alternatives.complementary results are given for £-consistent estimators havin~ a relatively large bias or variance under local alternatives.A simulation study illustrates the theoretical results of the second and the third chapter.The last chapter deals with power approximations for the Cressie-Read (1984) class of x 2 statistics when no nuisance parameters are present.Although classical (moment-corrected) x 2 approximations work reasonably well under the hypothesis, non-central x 2 power approximations are inadequate for moderate sample sizes.A non-local Taylor expansion of the test ii statistic yields a new approximation based on a weighted sum of independent 1 non-central x 2 distributions; the distribution error is of order O(n-2 ) uniformly in alternatives and levels.Exact power computations for n=20,50show that the new approximation is very good and is greatly superior to traditional ones.