Abstract
In Chap. 2, the applications were based on samples with complete data. In contrast, this chapter is devoted to presenting inferential procedures based on (mostly right) censored data. Heavy emphasis is given to the estimation of survival function since it plays an important role in the survival data analysis. Estimation procedures based on different priors and under various sampling schemes are discussed. Estimation of hazard rates and cumulative hazard functions is also included. This is followed by other examples which include estimation procedures in certain stochastic process models, Markov Chains, and competing risks models. Finally, estimation of the survival function in presence of covariates is presented.
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References
Blum, J., & Susarla, V. (1977). On the posterior distribution of a Dirichlet process given randomly right censored observations. Stochastic Processes and Their Applications, 5, 207–211.
Burridge, M. (1981). Empirical Bayes analysis of survival data. Journal of the Royal Statistical Society. Series B. Methodological, 43, 65–75.
Clayton, M. K. (1991). A Monte Carlo method for Bayesian inference in frailty models. Biometrika, 47, 467–485.
Damien, P., & Walker, S. (2002). A Bayesian nonparametric comparison of two treatments. Scandinavian Journal of Statistics, 29, 51–56.
Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Annals of Probability, 2, 183–201.
Dykstra, R. L., & Laud, P. (1981). A Bayesian nonparametric approach to reliability. Annals of Statistics, 9, 356–367.
Efron, B. (1967). The two sample problem with censored data. In Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, vol. 4: biology and problems of health.
Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230.
Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Annals of Statistics, 2, 615–629.
Ferguson, T. S., & Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Annals of Statistics, 7, 163–186.
Gardiner, J. C., & Susarla, V. (1981). A nonparametric estimator of the survival function under progressive censoring. In J. Crowley & R. A. Johnson (Eds.), IMS lecture notes—monograph series: Vol. 2. Survival analysis (pp. 26–40).
Gardiner, J. C., & Susarla, V. (1983). Weak convergence of a Bayesian nonparametric estimator of the survival function under progressive censoring. Statistics & Decisions, 1, 257–263.
Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52, 203–223.
Ghorai, J. K. (1981). Empirical Bayes estimation of a distribution function with a gamma process prior. Communications in Statistics. Theory and Methods, 10(12), 1239–1248.
Ghorai, J. K. (1989). Nonparametric Bayesian estimation of a survival function under the proportional hazard model. Communications in Statistics. Theory and Methods, 18(5), 1831–1842.
Gross, A. J., & Clark, V. A. (1975). Survival distributions. Reliability applications in biomedical sciences. New York: Wiley.
Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Annals of Statistics, 18(3), 1259–1294.
Hollander, M., & Korwar, R. M. (1976). Nonparametric empirical Bayes estimation of the probability that X≤Y. Communications in Statistics. Theory and Methods, 5(14), 1369–1383.
Johnson, N. L., & Kotz, S. (1970). Distributions in statistics—continuous multivariate distributions. New York: Wiley.
Johnson, R. A., Susarla, V., & Van Ryzin, J. (1979). Bayesian non-parametric estimation for age-dependent branching processes. Stochastic Processes and Their Applications, 9, 307–318.
Kalbfleisch, J. D. (1978). Nonparametric Bayesian analysis of survival data. Journal of the Royal Statistical Society. Series B. Methodological, 40, 214–221.
Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York: Wiley.
Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457–481.
Kim, Y. (1999). Nonparametric Bayesian estimators for counting processes. Annals of Statistics, 27, 562–588.
Lo, A. Y. (1981). Bayesian nonparametric statistical inference for shock models and wear processes. Scandinavian Journal of Statistics, 8, 237–242.
Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59, 55–66.
Lo, A. Y. (1993a). A Bayesian bootstrap for censored data. Annals of Statistics, 21, 100–123.
Lo, A. Y. (1993b). A Bayesian method for weighted sampling. Annals of Statistics, 21, 2138–2148.
Muliere, P., & Walker, S. (1997). A Bayesian non-parametric approach to survival analysis using Polya trees. Scandinavian Journal of Statistics, 24, 331–340.
Neath, A. A. (2003). Polya tree distributions for statistical modeling of censored data. Journal of Applied Mathematics & Decision Sciences, 7(3), 175–186.
Neath, A. A., & Samaniego, F. J. (1996). On Bayesian estimation of the multiple decrement function in the competing risks problem. Statistics & Probability Letters, 31, 75–83.
Neath, A. A., & Samaniego, F. J. (1997). On Bayesian estimation of the multiple decrement function in the competing risks problem, II. Statistics & Probability Letters, 35, 345–354.
Padgett, W. J., & Wei, L. J. (1981). A Bayesian nonparametric estimator of survival probability assuming increasing failure rate. Communications in Statistics. Theory and Methods, 10(1), 49–63.
Peterson, A. V. (1977). Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions. Journal of the American Statistical Association, 72, 854–858.
Phadia, E. G. (1980). A note on empirical Bayes estimation of a distribution function based on censored data. Annals of Statistics, 8(1), 226–229.
Phadia, E. G., & Susarla, V. (1983). Nonparametric Bayesian estimation of a survival curve with dependent censoring mechanism. Annals of the Institute of Statistical Mathematics, 35, 389–400.
Phadia, E. G., & Susarla, V. (1979). An empirical Bayes approach to two-sample problems with censored data. Communications in Statistics. Theory and Methods, 8(13), 1327–1351.
Ramsey, F. L. (1972). A Bayesian approach to bioassay. Biometrics, 28, 841–858.
Salinas-Torres, V. H., Pereira, C. A. B., & Tiwari, R. C. (2002). Bayesian nonparametric estimation in a series system or a competing-risks model. Journal of Nonparametric Statistics, 14, 449–458.
Samaniego, F. J., & Whitaker, L. R. (1988). On estimating population characteristics from record-breaking observations. II. Nonparametric results. Naval Research Logistics, 35, 221–236.
Sinha, D. (1998). Posterior likelihood methods for multivariate survival data. Biometrics, 54, 1463–1474.
Susarla, V., & Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations. Journal of the American Statistical Association, 71, 897–902.
Susarla, V., & Van Ryzin, J. (1978a). Empirical Bayes estimation of a distribution (survival) function from right-censored observations. Annals of Statistics, 6, 740–754.
Susarla, V., & Van Ryzin, J. (1978b). Large sample theory for a Bayesian nonparametric survival curve estimator based on censored samples. Annals of Statistics, 6, 755–768.
Susarla, V., & Van Ryzin, J. (1980). Addendum to “Large sample theory for a Bayesian nonparametric survival curve estimator based on censored samples”. Annals of Statistics, 8, 693.
Tiwari, R. C., & Zalkikar, J. N. (1991b). Bayesian inference of survival curve from record-breaking observations: estimation and asymptotic results. Naval Research Logistics, 38, 599–609.
Tiwari, R. C., & Zalkikar, J. N. (1993). Nonparametric Bayesian estimation of survival function under random left truncation. Journal of Statistical Planning and Inference, 35, 31–45.
Tiwari, R. C., Jammalamadaka, S. R., & Zalkikar, J. N. (1988). Bayes and empirical Bayes estimation of survival function under progressive censoring. Communications in Statistics. Theory and Methods, 17(10), 3591–3606.
Tsai, W. Y. (1986). Estimation of survival curves from dependent censorship models via a generalized self-consistent property with nonparametric Bayesian estimation application. Annals of Statistics, 14, 238–249.
Walker, S. G., & Muliere, P. (1997a). Beta-Stacy processes and a generalization of the Polya-urn scheme. Annals of Statistics, 25(4), 1762–1780.
Wild, C. J., & Kalbfleisch, J. D. (1981). A note on a paper by Ferguson and Phadia. Annals of Statistics, 9, 1061–1065.
Zalkikar, J. N., Tiwari, R. C., & Jammalamadaka, S. R. (1986). Bayes and empirical Bayes estimation of the probability that Z>X+Y. Communications in Statistics. Theory and Methods, 15(10), 3079–3101.
Zehnwirth, B. (1985). Nonparametric linear Bayes estimation of survival curves from incomplete observations. Communications in Statistics. Theory and Methods, 14(8), 1769–1778.
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Phadia, E.G. (2013). Inference Based on Incomplete Data. In: Prior Processes and Their Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39280-1_3
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DOI: https://doi.org/10.1007/978-3-642-39280-1_3
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