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Estimating a distribution function for censored time series data. (English) Zbl 1057.62068
Summary: Consider a long term study, where a series of dependent and possibly censored failure times is observed. Suppose that the failure times have a common marginal distribution function, but they exhibit a mode of time series structure such as $\alpha$-mixing. The inference on the marginal distribution function is of interest to us. The main results of this article show that, under some regularity conditions, the Kaplan-Meier estimator enjoys uniform consistency with rates, and a stochastic process generated by the Kaplan Meier estimator converges weakly to a certain Gaussian process with a specified covariance structure. Finally, an estimator of the limiting variance of the Kaplan-Meier estimator is proposed and its consistency is established.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62G07 Density estimation 60F05 Central limit and other weak theorems 62G05 Nonparametric estimation 62M09 Non-Markovian processes: estimation
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##### References:
 [1] Andrews, D. W. K.: Laws of large numbers for dependent nonidentically distributed random variables. Econ. theory 4, 458-467 (1988) [2] Auestad, B.; Tjøstheim, D.: Identification of nonlinear time series: first order characterization and order determination. Biometrika 77, 669-687 (1990) [3] Billingsley, P.: Convergence of probability measures. (1968) · Zbl 0172.21201 [4] Breslow, N.; Crowley, J.: A large sample study of the life table and product limit estimators under random censorship. Ann. statist. 2, 437-453 (1974) · Zbl 0283.62023 [5] Cai, Z.: Asymptotic properties of kaplan--meier estimator for censored dependent data. Statist. probab. Lett. 37, 381-389 (1998) · Zbl 0902.62040 [6] Cai, J.; Prentice, R. L.: Estimating equations for hazard ratio parameters based on correlated failure time data. Biometrika 82, 151-164 (1995) · Zbl 0823.62084 [7] Cai, Z.; Roussas, G. G.: Uniform strong estimation under ${\alpha}$-mixing, with rates. Statist. probab. Lett. 15, 47-55 (1992) · Zbl 0757.62024 [8] Cai, Z.; Roussas, G. G.: Kaplan--meier estimator under association. J. multivariate anal. 67, 318-348 (1998) · Zbl 1030.62521 [9] Chen, R.; Tsay, R.: Functional coefficient autoregressive models. J. amer. Statist. assoc. 88, 298-308 (1993) · Zbl 0776.62066 [10] Cox, R. D.; Oakes, D.: Analysis of survival data. (1984) [11] Gill, R. D.: Censoring and stochastic integrals. Mathematical centre tracts 124 (1980) · Zbl 0456.62003 [12] Gill, R. D.: Testing with replacement and the product limit estimator. Ann. statist. 9, 853-860 (1981) · Zbl 0468.62039 [13] Gill, R. D.: Large sample behavior of the product limit estimator on the whole line. Ann. statist. 11, 49-56 (1983) · Zbl 0518.62039 [14] Gorodetskii, V. V.: On the strong mixing property for linear sequences. Theory probab. Appl. 22, 411-413 (1977) [15] Hall, P.; Heyde, C. C.: Martingale limit theory and its applications. (1980) · Zbl 0462.60045 [16] Ibragimov, I. A.; Linnik, Yu.V.: Independent and stationary sequences of random variables. (1971) · Zbl 0219.60027 [17] Kang, S.; Koehler, K. J.: Modification of the greenwood formula for correlated response times. Biometrics 53, 885-899 (1997) · Zbl 0890.62081 [18] Kaplan, E. L.; Meier, P.: Nonparametric estimation from incomplete observations. J. amer. Statist. assoc. 53, 457-481 (1958) · Zbl 0089.14801 [19] Koehler, K. J.: An analysis of temperature effects on bean leaf egg hatch times. (1995) [20] Koehler, K. J.; Symanowski, J.: Constructing multivariate distributions with specific marginal distributions. J. multivariate anal. 55, 261-282 (1995) · Zbl 0863.62048 [21] N. N. Leonenko, and, L. M. Sakhno, On the Kaplan--Meier estimator of long-range dependent sequences, unpublished manuscript, 2000. · Zbl 1092.62510 [22] Masry, E.; Tjøstheim, D.: Nonparametric estimation and identification of nonlinear ARCH time series: strong convergence and asymptotic normality. Econ. theory 11, 258-289 (1995) [23] Masry, E.; Tjøstheim, D.: Additive nonlinear ARX time series and projection estimates. Econ. theory 13, 214-252 (1997) [24] Peterson, A. V.: Expressing the kaplan--meier estimator as a function of empirical subsurvival functions. J. amer. Statist. assoc. 72, 854-858 (1977) · Zbl 0372.62077 [25] Shao, Q. -M.; Yu, H.: Weak convergence for weighted empirical processes of dependent sequences. Ann. probab. 24, 2098-2127 (1996) · Zbl 0874.60006 [26] Shorack, G. R.; Wellner, J. A.: Empirical processes with applications to statistics. (1986) · Zbl 1170.62365 [27] Shumway, R. H.; Azari, A. S.; Johnson, P.: Estimating mean concentrations under transformation for environmental data with detection limits. Technometrics 31, 347-356 (1988) [28] Stute, W.; Wang, J. L.: A strong law under random censorship. Ann. statist. 21, 1591-1607 (1993) · Zbl 0785.60020 [29] Tucker, H.: A graduate course in probability. (1967) · Zbl 0159.45702 [30] Voelkel, J.; Crowley, J.: Nonparametric inference for a class of semi-Markov process with censored observation. Ann. statist. 12, 142-160 (1984) · Zbl 0552.62020 [31] Wang, J. G.: A note on the uniform consistency of the kaplan--meier estimator. Ann. statist. 15, 1313-1316 (1987) · Zbl 0631.62043 [32] Wei, L. J.; Lin, D. Y.; Weissfeld, L.: Regression analysis of multivariate incomplete failure times data by modeling marginal distributions. J. amer. Statist. assoc. 84, 1064-1072 (1989) [33] Withers, C. S.: Conditions for linear processes to be strong mixing. Z. wahrsch. Verw. gebiete 57, 477-480 (1981) · Zbl 0465.60032 [34] Ying, Z.; Wei, L. J.: The kaplan--meier estimate for dependent failure time observations. J. multivariate anal. 50, 17-29 (1994) · Zbl 0798.62048 [35] Yu, H.: A glivenko--cantelli lemma and weak convergence for empirical processes of associated sequences. Probab. theory related fields 95, 357-370 (1993) · Zbl 0792.60018