Empirical processes with estimated parameters under auxiliary information. (English) Zbl 1073.62044

Summary: Empirical processes with estimated parameters are a well established subject in nonparametric statistics. In the classical theory they are based on the empirical distribution function which is the nonparametric maximum likelihood estimator for a completely unknown distribution function. In the presence of some “nonparametric” auxiliary information about the distribution, like a known mean or a known median, for example, the nonparametric maximum likelihood estimator is a modified empirical distribution function which puts random masses on the observations in order to take the available information into account [see, e.g., A. B. Owen, Biometrika 75, 237–249 (1988; Zbl 0641.62032); Ann. Statist. 18, 90–120 (1990; Zbl 0712.62040); Empirical likelihood (2001; Zbl 0989.62019)]. B. Zhang [Metrika 46, 221–244 (1997; Zbl 0912.62058)] has proved a functional central limit theorem for the empirical process pertaining to this modified empirical distribution function.
We consider the corresponding empirical process with estimated parameters and derive its asymptotic distribution. The limiting process is a centered Gaussian process with a complicated covariance function depending on the unknown parameter. The result becomes useful in practice through the bootstrap, which is shown to be consistent in case of a known mean. The performance of the resulting bootstrap goodness-of-fit test based on the Kolmogorov - Smirnov statistic is studied through simulations.


62G30 Order statistics; empirical distribution functions
62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
60F17 Functional limit theorems; invariance principles
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