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A product‐limit estimator for use with length‐biased data
Author(s) -
Winter B. B.,
Földes A.
Publication year - 1988
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314932
Subject(s) - estimator , censoring (clinical trials) , mathematics , nonparametric statistics , statistics , consistency (knowledge bases) , population , cumulative distribution function , limit (mathematics) , discrete mathematics , probability density function , mathematical analysis , demography , sociology
Abstract The following life‐testing situation is considered. At some time in the distant past, n objects, from a population with life distribution F , were put in use; whenever an object failed, it was promptly replaced. At some time τ, long after the start of the process, a statistician starts observing the n objects in use at that time; he knows the age of each of those n objects, and observes each of them for a fixed length of time≪ ∞, or until failure, whichever occurs first. In the case where T is finite, some of the observations may be censored; in the case where T =∞, there is no censoring. The total life of an object in use at time ∞ is a length‐biased observation from F . A nonparametric estimator of the (cumulative) hazard function is proposed, and is used to construct an estimator of F which is of the product‐limit type. Strong uniform consistency results (for n → ∞) are obtained. An “Aalen‐Johansen” identity, satisfied by any pair of life distributions and their (cumulative) hazard functions, is used in obtaining rate‐of‐convergence results.