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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 n * , of E. L. Kaplan and 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 n o of D. Lynden-Bell [Mon. Not. R. Astron. Soc. 155, 95-118 (1971)], J. C. Jackson [ibid. 166, 281-295 (1974)], and 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 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 n * , F n o .

Reviewer: E.I.Trofimov
MSC:
62G05Nonparametric estimation
60F15Strong limit theorems
60F17Functional limit theorems; invariance principles
60H05Stochastic integrals
60B12Limit theorems for vector-valued random variables (infinite-dimensional case)
60G44Martingales with continuous parameter