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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.
62G20 Asymptotic properties of nonparametric inference
62N01 Censored data models
62H12 Estimation in multivariate analysis
62G07 Density estimation
62G05 Nonparametric estimation
Full Text: DOI
[1] Kim, M.Y.; Xue, X., The analysis of multivariate interval-censored survival data, Statistics in medicine, 21, 3715-3726, (2002)
[2] Goggins, W.B.; Finkelstein, D.M., A proportional hazards model for multivariate interval-censored failure time data, Biometrics, 56, 940-943, (2000) · Zbl 1060.62617
[3] Jones, G.; Rocke, D.M., Multivariate survival analysis with doubly-censored data: application to the assessment of accutane treatment for fibrodysplasia ossificans progressiva, Statistics in medicine, 21, 2547-2562, (2002)
[4] Wong, G.Y.C.; Yu, Q.Q., Generalized MLE of a joint distribution function with multivariate interval-censored data, Journal of multivariate analysis, 69, 155-166, (1999) · Zbl 0931.62084
[5] Yu, S.H.; Yu, Q.Q.; Wong, G.Y.C., Consistency of the generalized MLE of a joint distribution function with multivariate interval-censored data, Journal of multivariate analysis, 97, 720-732, (2006) · Zbl 1333.62129
[6] DeGruttola, V.; Lagakos, S.W., Analysis of doubly-censored survival data, with application to AIDS, Biometrics, 45, 1-11, (1989) · Zbl 0715.62223
[7] Gómez, G.; Calle, M.L., Nonparametric estimation with doubly censored data, Journal of applied statistics, 26, 45-58, (1999) · Zbl 1072.62553
[8] Gómez, G.; Lagakos, S.W., Estimation of the infection time and latency distribution of AIDS with doubly censored data, Biometrics, 50, 204-212, (1994) · Zbl 0826.62088
[9] Sun, J., Empirical estimation of a distribution function with truncated and doubly interval-censored data and its application to AIDS studies, Biometrics, 51, 1094-1104, (1995) · Zbl 0875.62497
[10] Sun, J., Self-consistency estimation of distributions based on truncated and doubly interval-censored data with applications to AIDS cohort studies, Lifetime data analysis, 3, 4, 305-313, (1997) · Zbl 1089.62534
[11] Fang, H.; Sun, J., Consistency of nonparametric maximum likelihood estimation of a distribution function based on doubly interval-censored failure time data, Statistics and probability letters, 55, 311-318, (2001) · Zbl 0994.62024
[12] Komárek, A.; Lesaffre, E., Bayesian semi-parametric accelerated failure tme model for paired doubly interval-censored data, Statistical modelling, 6, 3-22, (2006)
[13] Komárek, A.; Lesaffre, E., Bayesian accelerated failure time model with multivariate doubly interval-censored data and flexible distributional assumptions, Journal of the American statistical association, 103, 523-533, (2008) · Zbl 05564507
[14] Komárek, A.; Lesaffre, E.; Härkänen, T.; Declerck, D.; Virtanen, I.J., A Bayesian analysis of multivariate doubly-interval-censored data, Biostatistics, 6, 145-155, (2005) · Zbl 1069.62099
[15] Groeneboom, P., Lectures on inverse problems, () · Zbl 0907.62042
[16] D. Deng, H. Fang, J. Sun, Nonparametric estimation for doubly censored failure time data, Journal of Nonparametric Statistics (2008) (submitted for publication) · Zbl 1172.62006
[17] Groeneboom, P.; Wellner, J.A., Information bounds and nonparametric maximum likelihood estimation, (1992), Birkhäuser Verlag Basel · Zbl 0757.62017
[18] van De Geer, S., Rates of convergence for the maximum likelihood estimator in mixture models, Journal of nonparametric statistics, 6, 293-310, (1996) · Zbl 0872.62039
[19] Deng, D.; Fang, H., On nonparametric maximum likelihood estimations of multivariate distribution function based on interval-censored data, Communications in statistics - theory and methods, 38, 54-74, (2008) · Zbl 1292.62074
[20] Geskus, R.B.; Groeneboom, P., Asymptotically optional estimation of smooth functionals for interval censoring, part 2, Statistica neerlandica, 51, 201-219, (1997) · Zbl 0891.62033
[21] Geskus, R.B.; Groeneboom, P., Asymptotically optional estimation of smooth functional for interval censoring, case 2, The annals of statistics, 27, 627-674, (1999) · Zbl 0954.62034
[22] Betensky, R.A.; Finkelstein, D.M., A non-parametric maximum likelihood estimator for bivariate interval censored data, Statistics in medicine, 18, 3089-3100, (1999)
[23] Gentleman, R.; Vandal, A.C., Computational algorithms for censored data problems using intersection graphs, Journal of computational & graphical statistics, 10, 403-421, (2001)
[24] Gentleman, R.; Vandal, A.C., Nonparametric estimation of the bivariate CDF for arbitrarily censored data, The Canadian journal of statistics, 30, 557-571, (2002) · Zbl 1018.62022
[25] Bogaerts, K.; Lesaffre, E., A new, fast algorithm to find the regions of possible support for bivariate interval-censored data, Journal of computational graphical statistics, 13, 330-340, (2004)
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