Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times.

*(English)*Zbl 1163.62037Summary: 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.

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 |

##### Keywords:

multivariate doubly interval-censored; nonparametric maximum likelihood estimation; strong consistency; convergence rate; epidemiology
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\textit{D. Deng} and \textit{H.-B. Fang}, J. Multivariate Anal. 100, No. 8, 1802--1815 (2009; Zbl 1163.62037)

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