Skip to main content

Inference Based on Incomplete Data

  • Chapter
Prior Processes and Their Applications
  • 1575 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

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.

    Article  MathSciNet  MATH  Google Scholar 

  • Burridge, M. (1981). Empirical Bayes analysis of survival data. Journal of the Royal Statistical Society. Series B. Methodological, 43, 65–75.

    MathSciNet  MATH  Google Scholar 

  • Clayton, M. K. (1991). A Monte Carlo method for Bayesian inference in frailty models. Biometrika, 47, 467–485.

    Google Scholar 

  • Damien, P., & Walker, S. (2002). A Bayesian nonparametric comparison of two treatments. Scandinavian Journal of Statistics, 29, 51–56.

    Article  MathSciNet  MATH  Google Scholar 

  • Doksum, K. A. (1974). Tailfree and neutral random probabilities and their posterior distributions. Annals of Probability, 2, 183–201.

    Article  MathSciNet  MATH  Google Scholar 

  • Dykstra, R. L., & Laud, P. (1981). A Bayesian nonparametric approach to reliability. Annals of Statistics, 9, 356–367.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Google Scholar 

  • Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferguson, T. S. (1974). Prior distributions on spaces of probability measures. Annals of Statistics, 2, 615–629.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferguson, T. S., & Phadia, E. G. (1979). Bayesian nonparametric estimation based on censored data. Annals of Statistics, 7, 163–186.

    Article  MathSciNet  MATH  Google Scholar 

  • 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).

    Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily singly-censored samples. Biometrika, 52, 203–223.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Gross, A. J., & Clark, V. A. (1975). Survival distributions. Reliability applications in biomedical sciences. New York: Wiley.

    MATH  Google Scholar 

  • Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Annals of Statistics, 18(3), 1259–1294.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Johnson, N. L., & Kotz, S. (1970). Distributions in statistics—continuous multivariate distributions. New York: Wiley.

    Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Kalbfleisch, J. D. (1978). Nonparametric Bayesian analysis of survival data. Journal of the Royal Statistical Society. Series B. Methodological, 40, 214–221.

    MathSciNet  MATH  Google Scholar 

  • Kalbfleisch, J. D., & Prentice, R. L. (1980). The statistical analysis of failure time data. New York: Wiley.

    MATH  Google Scholar 

  • Kaplan, E. L., & Meier, P. (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457–481.

    Article  MathSciNet  MATH  Google Scholar 

  • Kim, Y. (1999). Nonparametric Bayesian estimators for counting processes. Annals of Statistics, 27, 562–588.

    Article  MathSciNet  MATH  Google Scholar 

  • Lo, A. Y. (1981). Bayesian nonparametric statistical inference for shock models and wear processes. Scandinavian Journal of Statistics, 8, 237–242.

    MATH  Google Scholar 

  • Lo, A. Y. (1982). Bayesian nonparametric statistical inference for Poisson point processes. Zeitschrift Für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 59, 55–66.

    Article  MATH  Google Scholar 

  • Lo, A. Y. (1993a). A Bayesian bootstrap for censored data. Annals of Statistics, 21, 100–123.

    Article  MathSciNet  MATH  Google Scholar 

  • Lo, A. Y. (1993b). A Bayesian method for weighted sampling. Annals of Statistics, 21, 2138–2148.

    Article  MathSciNet  MATH  Google Scholar 

  • Muliere, P., & Walker, S. (1997). A Bayesian non-parametric approach to survival analysis using Polya trees. Scandinavian Journal of Statistics, 24, 331–340.

    Article  MathSciNet  MATH  Google Scholar 

  • Neath, A. A. (2003). Polya tree distributions for statistical modeling of censored data. Journal of Applied Mathematics & Decision Sciences, 7(3), 175–186.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Ramsey, F. L. (1972). A Bayesian approach to bioassay. Biometrics, 28, 841–858.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Samaniego, F. J., & Whitaker, L. R. (1988). On estimating population characteristics from record-breaking observations. II. Nonparametric results. Naval Research Logistics, 35, 221–236.

    Article  MathSciNet  MATH  Google Scholar 

  • Sinha, D. (1998). Posterior likelihood methods for multivariate survival data. Biometrics, 54, 1463–1474.

    Article  MATH  Google Scholar 

  • Susarla, V., & Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curves from incomplete observations. Journal of the American Statistical Association, 71, 897–902.

    Article  MathSciNet  MATH  Google Scholar 

  • Susarla, V., & Van Ryzin, J. (1978a). Empirical Bayes estimation of a distribution (survival) function from right-censored observations. Annals of Statistics, 6, 740–754.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Walker, S. G., & Muliere, P. (1997a). Beta-Stacy processes and a generalization of the Polya-urn scheme. Annals of Statistics, 25(4), 1762–1780.

    Article  MathSciNet  MATH  Google Scholar 

  • Wild, C. J., & Kalbfleisch, J. D. (1981). A note on a paper by Ferguson and Phadia. Annals of Statistics, 9, 1061–1065.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Zehnwirth, B. (1985). Nonparametric linear Bayes estimation of survival curves from incomplete observations. Communications in Statistics. Theory and Methods, 14(8), 1769–1778.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

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

Download citation

Publish with us

Policies and ethics