×
Parner, Erik

Asymptotic theory for the correlated gamma-frailty model. (English) Zbl 0934.62101

Ann. Stat. 26, No. 1, 183-214 (1998).
Summary: The frailty model is a generalization of Cox’s proportional hazard model, where a shared unobserved quantity in the intensity induces a positive correlation among the survival times. S. A. Murphy [ibid. 22, No. 2, 712-731 (1994; Zbl 0827.62033); ibid. 23, No. 1, 182-198 (1995; Zbl 0822.62069)] showed consistency and asymptotic normality of the nonparametric maximum likelihood estimator (NPMLE) for the shared gamma-frailty model without covariates. We extend this result to the correlated gamma-frailty model, and we allow for covariates.
We discuss the definition of the nonparametric likelihood function in terms of a classical proof of consistency for the maximum likelihood estimator, which goes back to A. Wald [Ann. Math. Statistics, Baltimore Md. 20, 595-601 (1949; Zbl 0034.22902)]. Our proof of the consistency for the NPMLE is essentially the same as the classical proof for the maximum likelihood estimator. A new central limit theorem for processes of bounded variation is given. Furthermore, we prove that a consistent estimator for the asymptotic variance of the NPMLE is given by the inverse of a discrete observed information matrix.

Cited in 107 Documents

MSC:

62N02 Estimation in survival analysis and censored data
62M09 Non-Markovian processes: estimation
62G20 Asymptotic properties of nonparametric inference
62G05 Nonparametric estimation

Keywords:

survival data; heterogeneity; correlated frailty; semiparametric models; nonparametric maximum likelihood estimator; central limit theorem

Citations:

Zbl 0827.62033; Zbl 0822.62069; Zbl 0034.22902
Cite Review PDF
Full Text: DOI

References:

[1] ANDERSEN, P. K., BORGAN, Ø., GILL, R. D. and KEIDING, N. 1993. Statistical Models Based on Counting Processes. Springer, Berlin. Z. · Zbl 0769.62061
[2] ANDERSEN, P. K. and GILL, R. D. 1982. Cox’s regression model for counting processes: a large sample study. Ann. Statist. 10 1100 1120. Z. · Zbl 0526.62026 · doi:10.1214/aos/1176345976
[3] BICKEL, P. 1985. Efficient testing in a class of transformation models. Bull. Int. Statist. Inst. 51 63 81, Meeting 23. Amsterdam. Z.
[4] BICKEL, P., KLAASSEN, C., RITOV, Y. and WELLNER, J. 1993. Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press. Z. · Zbl 0786.62001
[5] BREMAUD, P. 1981. Point Processes and Queues. Springer, New York. Z. · Zbl 0478.60004
[6] CLAy TON, D. G. and CUZICK, J. 1985. Multivariate generalizations of the proportional hazards Z. model with discussion. J. Roy. Statist. Soc. Ser. A 148 82 117. Z. JSTOR: · Zbl 0581.62086 · doi:10.2307/2981943
[7] DABROWSKA, D. M. 1988. Kaplan Meier estimate on the plane. Ann. Statist. 16 1475 1489. Z. · Zbl 0653.62071 · doi:10.1214/aos/1176351049
[8] ELBERS, C. and RIDDER, G. 1982. True and spurious duration dependence: the identifiability of the proportional hazard model. Rev. Econom. Stud. XLIX 403 409. Z. JSTOR: · Zbl 0512.62113 · doi:10.2307/2297364
[9] GILL, R. D. 1985. Discussion of the paper by D. Clay ton and J. Cuzick. J. Roy. Statist. Soc. Ser. A 148 108 109.
[10] GILL, R. D. 1989. Nonand semi-parametrix maximum likelihood estimation and the von Mises method, 1. Scand. J. Statist. 16 97 128. Z.
[11] GILL, R. D. 1992. Marginal partial likelihood. Scand. J. Statist. 19 133 137. Z. · Zbl 0747.62008
[12] GROENEBOOM, P. 1991. Nonparametric maximum likelihood estimators for interval censoring and deconvolution. Technical Report 378, Dept. Statistics, Stanford Univ. Z.
[13] HILDEBRANDT, T. H. 1963. Introduction to the Theory of Integration. Academic Press, New York. Z. HOFFMANN-JøRGENSEN, J. 1994. Probability with a View to Statistics. Chapman and Hall, London. Z. · Zbl 0112.28302
[14] HOUGAARD, P. 1987. Modelling multivariate survival. Scand. J. Statist. 14 291 304. Z. · Zbl 0644.62099
[15] JOHANSEN, S. 1983. An extension of Cox’s regression model. Internat. Statist. Rev. 51 258 262. Z. JSTOR: · Zbl 0526.62081 · doi:10.2307/1402746
[16] KIEFER, J. and WOLFOWITZ, J. 1956. Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Statist. 27 887 906. Z. · Zbl 0073.14701 · doi:10.1214/aoms/1177728066
[17] KORTRAM, R. A., VAN ROOIJ, A. C. M., LENSTRA, A. J. and RIDDER, G. 1995. Constructive identification of the mixed proportional hazards model. Statist. Neerlandica 49 269 281. Z. · Zbl 0840.62051 · doi:10.1111/j.1467-9574.1995.tb01469.x
[18] MORSING, T. 1994. Competing risks in cross-over designs. Dept. Mathematics, Chalmers Univ. Technology. Preprint 20. Z.
[19] MURPHY, S. A. 1994. Consistency in a proportional hazard model incorporating a random effect. Ann. Statist. 22 712 731. Z. · Zbl 0827.62033 · doi:10.1214/aos/1176325492
[20] MURPHY, S. A. 1995. Asy mptotic theory for the frailty model. Ann. Statist. 23 182 198. Z. · Zbl 0822.62069 · doi:10.1214/aos/1176324462
[21] NIELSEN, G. G., GILL, R. D., ANDERSEN, P. K. and SøRENSEN, T. I. A. 1992. A counting process approach to maximum likelihood estimation in frailty models. Scand. J. Statist. 19 25 44. Z. · Zbl 0747.62093
[22] PARNER, E. 1996a. Consistency in the correlated Gamma-frailty model. Research Report 345, Dept. Theoretical Statistics, Inst. Mathematics, Univ. Aarhus. Z. JSTOR: · Zbl 0984.62089 · doi:10.1093/biomet/88.2.459
[23] PARNER, E. 1996b. Asy mptotic normality in the correlated Gamma-frailty model. Research Report 346, Dept. Theoretical Statistics, Inst. Mathematics, Univ. Aarhus. Z. JSTOR: · Zbl 0984.62089 · doi:10.1093/biomet/88.2.459
[24] PEDERSEN, J. 1995. A litter frailty model. Research Report 95 13, Dept. Biostatistics, Univ. Copenhagen. Z.
[25] RAO, R. R. 1963. The law of large numbers for D 0, 1 -valued random variables. Theory Probab. Appl. 8 70 74. Z. · Zbl 0122.13303 · doi:10.1137/1108005
[26] RUDIN, W. 1973. Functional Analy sis. McGraw-Hill, New York. Z.
[27] VAN DER VAART, A. W. 1995. Efficiency of infinitely dimensional estimators. Statist. Neerlandica 49 9 30. Z. · Zbl 0830.62029 · doi:10.1111/j.1467-9574.1995.tb01452.x
[28] VAN DER VAART, A. W. and WELLNER, J. A. 1996. Weak Convergence and Empirical Processes. Springer, New York. Z. · Zbl 0862.60002
[29] VAUPEL, J. W., MANTON, K. G. and STALLARD, E. 1979. The impact of heterogeneity in individual frailty on the dy namics of mortality. Demography 16 439 454. Z.
[30] WALD, A. 1949. Note on the consistency of maximum likelihood estimate. Ann. Math. Statist. 20 595 601. Z. · Zbl 0034.22902 · doi:10.1214/aoms/1177729952
[31] WIJERS, B. J. 1995. Consistent non-parametric estimation for a one-dimensional line segment process observed in an interval. Scand. J. Statist. 22 335 360. Z. · Zbl 0844.62027
[32] WOODBURY, M. A. 1971. Discussion of paper by Hartley and Hocking. Biometrics 27 808 817. Z.
[33] YASHIN, A., VAUPEL, J. and IACHINE, I. 1995. Correlated individual frailty: an advantageous approach to survival analysis of bivariate data. Mathematical Population Studies. · Zbl 0873.62121
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.