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Backward estimation of stochastic processes with failure events as time origins. (English) Zbl 1202.62108

Summary: Stochastic processes often exhibit sudden systematic changes in pattern a short time before certain failure events. Examples include increase in medical costs before death and decrease in CD4 counts before AIDS diagnosis. To study such terminal behavior of stochastic processes, a natural and direct way is to align the processes using failure events as time origins. This paper studies backward stochastic processes counting time backward from failure events, and proposes one-sample nonparametric estimation of the mean of backward processes when follow-up is subject to left truncation and right censoring. We will discuss benefits of including prevalent cohort data to enlarge the identifiable region and large sample properties of the proposed estimator with related extensions. A SEER-Medicare linked data set is used to illustrate the proposed methodologies.

MSC:

62M09 Non-Markovian processes: estimation
62N02 Estimation in survival analysis and censored data
92C50 Medical applications (general)
62P10 Applications of statistics to biology and medical sciences; meta analysis
62G05 Nonparametric estimation
62N01 Censored data models
65C60 Computational problems in statistics (MSC2010)

References:

[1] Bang, H. and Tsiatis, A. A. (2000). Estimating medical costs with censored data. Biometrika 87 329-343. · Zbl 0963.62094 · doi:10.1093/biomet/87.2.329
[2] Bilias, Y., Gu, M. and Ying, Z. (1997). Towards a general asymptotic theory for Cox model with staggered entry. Ann. Statist. 25 662-682. · Zbl 0923.62085 · doi:10.1214/aos/1031833668
[3] Chan, I. S. F., Neaton, J. D., Saravolatz, L. D., Crane, L. R. and Osterberger, J. (1995). Frequencies of opportunistic diseases prior to death among HIV-infected persons. Aids 9 1145-1151.
[4] Cook, R. J. and Lawless, J. F. (1997). Marginal analysis of recurrent events and a terminating event. Stat. Med. 16 911-924.
[5] Ghosh, D. and Lin, D. Y. (2000). Nonparametric analysis of recurrent events and death. Biometrics 56 554-562. · Zbl 1060.62614 · doi:10.1111/j.0006-341X.2000.00554.x
[6] Gross, S. T. and Lai, T. L. (1996). Nonparametric estimation and regression analysis with left-truncated and right-censored data. J. Amer. Statist. Assoc. 91 1166-1180. · Zbl 0882.62037 · doi:10.2307/2291735
[7] Huang, Y. and Louis, T. A. (1998). Nonparametric estimation of the joint distribution of survival time and mark variables. Biometrika 85 785-798. · Zbl 0921.62035 · doi:10.1093/biomet/85.4.785
[8] Lai, T. L. and Ying, Z. (1991). Estimating a distribution function with truncated and censored data. Ann. Statist. 19 417-442. · Zbl 0741.62037 · doi:10.1214/aos/1176347991
[9] Lawless, J. F. and Nadeau, C. (1995). Some simple robust methods for the analysis of recurrent events. Technometrics 37 158-168. · Zbl 0822.62085 · doi:10.2307/1269617
[10] Lin, D. Y. (2000). Proportional means regression for censored medical costs. Biometrics 56 775-778. · Zbl 1060.62635 · doi:10.1111/j.0006-341X.2000.00775.x
[11] Lin, D. Y., Fleming, T. R. and Wei, L. J. (1994). Confidence bands for survival curves under the proportional hazards model. Biometrika 81 73-81. · Zbl 0800.62708 · doi:10.2307/2337051
[12] Lin, D. Y., Feuer, E. J., Etzioni, R. and Wax, Y. (1997). Estimating medical costs from incomplete follow-up data. Biometrics 53 419-434. · Zbl 0881.62116 · doi:10.2307/2533947
[13] Lin, D. Y., Wei, L. J., Yang, I. and Ying, Z. (2000). Semiparametric regression for the mean and rate functions of recurrent events. J. Roy. Statist. Soc. Ser. B Stat. Methodol. 62 711-730. · Zbl 1074.62510 · doi:10.1111/1467-9868.00259
[14] Nelson, W. (1988). Graphical analysis of system repair data. Journal of Quality Technology 20 24-35.
[15] Pawitan, Y. and Self, S. (1993). Modeling disease marker processes in AIDS. J. Amer. Statist. Assoc. 88 719-726. · Zbl 0800.62728 · doi:10.2307/2290756
[16] Pepe, M. S. and Cai, J. (1993). Some graphical displays and marginal regression analyses for recurrent failure times and time dependent covariates. J. Amer. Statist. Assoc. 88 811-820. · Zbl 0794.62074 · doi:10.2307/2290770
[17] Pollard, D. (1990). Empirical Processes: Theory and Applications . IMS, Hayward, CA. · Zbl 0741.60001
[18] Strawderman, R. L. (2000). Estimating the mean of an increasing stochastic process at a censored stopping time. J. Amer. Statist. Assoc. 95 . · Zbl 1008.62098 · doi:10.2307/2669760
[19] Tsai, W.-Y., Jewell, N. P. and Wang, M.-C. (1987). A note on the product-limit estimator under right censoring and left truncation. Biometrika 74 883-886. · Zbl 0628.62101 · doi:10.1093/biomet/74.4.883
[20] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics . Springer, New York. · Zbl 0862.60002
[21] Wang, M.-C. (1991). Nonparametric estimation from cross-sectional survival data. J. Amer. Statist. Assoc. 86 130-143. · Zbl 0739.62026 · doi:10.2307/2289722
[22] Wang, M.-C. and Chiang, C. T. (2002). Non-parametric methods for recurrent event data with informative and non-informative censorings. Stat. Med. 21 445-456.
[23] Wang, M.-C., Qin, J. and Chiang, C. T. (2001). Analyzing recurrent event data with informative censoring. J. Amer. Statist. Assoc. 96 1057-1065. · Zbl 1072.62646 · doi:10.1198/016214501753209031
[24] Warren, J. L., Klabunde, C. N., Schrag, D., Bach, P. B. and Riley, G. F. (2002). Overview of the SEER-Medicare data: Content, research applications, and generalizability to the United States elderly population. Med. Care 40 3-18.
[25] Woodroofe, M. (1985). Estimating a distribution function with truncated data. Ann. Statist. 13 163-177. · Zbl 0574.62040 · doi:10.1214/aos/1176346584
[26] Zhao, H. and Tian, L. (2001). On estimating medical cost and incremental cost-effectiveness ratios with censored data. Biometrics 57 1002-1008. · Zbl 1209.62346 · doi:10.1111/j.0006-341X.2001.01002.x
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