Summary
Let ℱ={F θ} be a parametric family of distribution functions, and denote withF n the empirical d.f. of an i.i.d. sample. Goodness-of-fit tests of a composite hypothesis (contained in ℱ) are usually based on the so-called estimated empirical process. Typically, they are not distribution-free. In such a situation the bootstrap offers a useful alternative. It is the purpose of this paper to show that this approximation holds with probability one. A simulation study is included which demonstrates the validity of the bootstrap for several selected parametric families.
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References
Bickel PJ, Freedman DA (1981) Some asymptotic theory for the bootstrap. Ann Statist 9:1196–1217
Chandra M, Singpurwalla ND and Stephens MA (1981) Kolmogorov statistics for tests of fit for the extreme value and Weibull distributions. JASA 76:719–731
D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Marcel Dekker, New York
Devroye L (1986) Non-Uniform random variate generation. Springer, New York- Berlin - Heidelberg
Durbin J, Knott M (1972) Components of Cramér-von Mises statistics. I J Roy Statist Soc B 34:290–307
Durbin J (1973) Weak convergence of the sample distribution function when parameters are estimated. Ann Statist 1:279–290
Durbin J, Knott M and Taylor CC (1975) Components of Cramér-von Mises statistics. II J Roy Statist Soc B 37:216–237
Lilliefors HW (1967) On the Kolmogorov-Smirnov test for normality with mean and variance unknown. JASA 62:399–402
Lilliefors HW (1969) On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. JASA 64:387–389
Pollard D (1984) Convergence of stochastic processes. Springer, New York- Berlin - Heidelberg
Seber GAF (1984) Multivariate observation. Wiley, New York
Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York
Stephens MA (1974) EDF statistics for goodness-of-fit and some comparisons. JASA 69:730–737
Stephens MA (1976) Asymptotic results for goodness-of-fit statistics with unknown parameters. Ann Statist 4:357–369
Stephens MA (1977) Goodness-of-fit for the extreme value distribution. Biometrika 64:583–588
Stephens MA (1979) Tests of fit for the logistic distribution based on the empirical distribution function. Biometrika 66:591–595
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Stute, W., Manteiga, W.G. & Quindimil, M.P. Bootstrap based goodness-of-fit-tests. Metrika 40, 243–256 (1993). https://doi.org/10.1007/BF02613687
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DOI: https://doi.org/10.1007/BF02613687