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Galen R. Shorack 《Revue canadienne de statistique》1984,12(4):319-332
Empirical and rank processes of a sample are shown to converge under weak conditions. These results are then extended to analogous empirical and rank processes formed from the residuals of a linear model. Representations of the limiting processes are obtained by means of a special construction. The technicalities are honed to require few regularity conditions. 相似文献
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Galen R. Shorack 《统计学通讯:理论与方法》2013,42(9):961-972
The bootstrap principle is justified for. robust M-estimates in regression, (A short proof justifying bootstrapping the empirical process is also given.) 相似文献
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We consider Z±n= sup0< t ≤ 1/22 U±n (t)/(t(1- t))1/2, where + and -denote the positive and negative parts respectively of the sample paths of the empirical process Un. U±n and Un are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z±n/log2n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z±n we show that Z-n/(2 log2n)1/2 is the appropriate normalization for Z-n. Csörgö & Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by -the sup over [an, 1-an] where an→0, how fast can an→0 and still have |Zn|/(2 log2n)1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |Un(t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest. 相似文献
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