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Functional laws of the iterated logarithm for the product-limit estimator of a distribution function under random censorship or truncation. (English) Zbl 0705.62040
The authors study the maximum likelihood estimators $F\sp*\sb n$, of {\it E. L. Kaplan} and {\it P. Meier} [J. Am. Stat. Assoc. 53, 457-481 (1958; Zbl 0089.148)], for a continuous distribution function F onto R, on censored independent random variables, and those for $F\sp o\sb n$ of {\it D. Lynden-Bell} [Mon. Not. R. Astron. Soc. 155, 95-118 (1971)], {\it J. C. Jackson} [ibid. 166, 281-295 (1974)], and {\it M. Woodroofe} [Ann. Stat. 13, 163-177 (1985; Zbl 0574.62040)], on truncated data, and their regularisations near the boundary of the support supp(dF). The theorems develop ideas of {\it R. Gill} [ibid. 11, 49-58 (1983; Zbl 0518.62039)] and give the compact LIL and the Glivenko-Cantelli like convergence with a power rate for these estimators in the space of right continuous functions D(supp(dF)), endowed with the uniform norm. The proofs are based on known martingale representations and martingale functional CLTs for $F\sp*\sb n$, $F\sp o\sb n$.
Reviewer: E.I.Trofimov

##### MSC:
 62G05 Nonparametric estimation 60F15 Strong limit theorems 60F17 Functional limit theorems; invariance principles 60H05 Stochastic integrals 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60G44 Martingales with continuous parameter
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