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Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times. (English) Zbl 1163.62037
Summary: This paper considers nonparametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of the multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer from censoring. As a consequence, the estimation of the multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown.
We investigate both the consistency and the convergence rate of a commonly used nonparametric estimate and show that as the dimension of the multivariate failure time increases or the number of censoring intervals of the multivariate failure time decreases, the convergence rate for the nonparametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.
MSC:
62G20 Asymptotic properties of nonparametric inference
62N01 Censored data models
62H12 Estimation in multivariate analysis
62G07 Density estimation
62G05 Nonparametric estimation
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