zbMATH — the first resource for mathematics

Bootstrap based goodness-of-fit-tests. (English) Zbl 0770.62016
Summary: Let \({\mathcal F}=\{F_ \theta\}\) be a parametric family of distribution functions, and denote with \(F_ n\) the empirical d.f. of an i.i.d. sample. Goodness-of-fit tests of a composite hypothesis (contained in \(\mathcal F\)) 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.

62F05 Asymptotic properties of parametric tests
62F03 Parametric hypothesis testing
Full Text: DOI EuDML
[1] Bickel PJ, Freedman DA (1981) Some asymptotic theory for the bootstrap. Ann Statist 9:1196–1217 · Zbl 0472.62054 · doi:10.1214/aos/1176345637
[2] 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
[3] D’Agostino RB, Stephens MA (1986) Goodness-of-fit techniques. Marcel Dekker, New York
[4] Devroye L (1986) Non-Uniform random variate generation. Springer, New York- Berlin - Heidelberg · Zbl 0593.65005
[5] Durbin J, Knott M (1972) Components of Cramér-von Mises statistics. I J Roy Statist Soc B 34:290–307 · Zbl 0238.62052
[6] Durbin J (1973) Weak convergence of the sample distribution function when parameters are estimated. Ann Statist 1:279–290 · Zbl 0256.62021 · doi:10.1214/aos/1176342365
[7] Durbin J, Knott M and Taylor CC (1975) Components of Cramér-von Mises statistics. II J Roy Statist Soc B 37:216–237 · Zbl 0335.62032
[8] Lilliefors HW (1967) On the Kolmogorov-Smirnov test for normality with mean and variance unknown. JASA 62:399–402
[9] Lilliefors HW (1969) On the Kolmogorov-Smirnov test for the exponential distribution with mean unknown. JASA 64:387–389
[10] Pollard D (1984) Convergence of stochastic processes. Springer, New York- Berlin - Heidelberg · Zbl 0544.60045
[11] Seber GAF (1984) Multivariate observation. Wiley, New York
[12] Serfling RJ (1980) Approximation theorems of mathematical statistics. Wiley, New York · Zbl 0538.62002
[13] Stephens MA (1974) EDF statistics for goodness-of-fit and some comparisons. JASA 69:730–737
[14] Stephens MA (1976) Asymptotic results for goodness-of-fit statistics with unknown parameters. Ann Statist 4:357–369 · Zbl 0325.62014 · doi:10.1214/aos/1176343411
[15] Stephens MA (1977) Goodness-of-fit for the extreme value distribution. Biometrika 64:583–588 · Zbl 0374.62028 · doi:10.1093/biomet/64.3.583
[16] Stephens MA (1979) Tests of fit for the logistic distribution based on the empirical distribution function. Biometrika 66:591–595 · Zbl 0417.62035 · doi:10.1093/biomet/66.3.591
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.