# Weak Convergence for Nonparametric Bayes Estimators Based on Beta Processes in the Random Censorship Model

Hong, Jee-Chang

• Published : 2005.12.01
• 46 9

#### Abstract

Hjort(1990) obtained the nonparametric Bayes estimator $\^{F}_{c,a}$ of $F_0$ with respect to beta processes in the random censorship model. Let $X_1,{\cdots},X_n$ be i.i.d. $F_0$ and let $C_1,{\cdot},\;C_n$ be i.i.d. G. Assume that $F_0$ and G are continuous. This paper shows that {$\^{F}_{c,a}$(u){\|}0 < u < T} converges weakly to a Gaussian process whenever T < $\infty$ and $\~{F}_0({\tau})\;<\;1$.

#### Keywords

Nonparametric Bayes estimator;Compact differentiability;Delta method

#### References

1. Billingsley, P. (1986). Probability and Measure. John Wiley & Son. New York
2. Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probability 2 183-201 https://doi.org/10.1214/aop/1176996703
3. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. statist. 1 209-230 https://doi.org/10.1214/aos/1176342360
4. Ferguson, T. S. and Phadia, E. G. (1979). Bayesian nonparametric estimation based one censored data. Ann. Statist. 7 163-186 https://doi.org/10.1214/aos/1176344562
5. Gill, R. D. (1989). Non- and Semi-parametric maximum likelihood estimators and the von Mises method (Part I). Scand. J. Statist. 16 97-128
6. Gill, R. D. (994). Lectures on probability theory. Springer-Verlag. New York
7. Gill, R. D. and Johansen, S. (1990). A survey of product-integration with a view toward application in survival analysis. Ann. Statist. 18 1501-1555 https://doi.org/10.1214/aos/1176347865
8. Hjort, N. L. (1990). Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18 1259-1294 https://doi.org/10.1214/aos/1176347749
9. Susarla, V. and Van Ryzin, J. (1976). Nonparametric Bayesian estimation of survival curve from incomplete observations. J. Amer. Statist. Assoc. 61 897-902
10. Susarla, V. and Van Ryzin, J. (1978). Large sample theory for a Bayesian nonparametric survival curve estimator based on censored samples. Ann. Statist. 6 755-768 https://doi.org/10.1214/aos/1176344250
11. Van der Vaart, A. W. and Wellner, J. A. (1996). Weak convergence and Empirical processes. Springer-Verlag. New York