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Local likelihood and local partial likelihood in hazard regression. (English) Zbl 0890.62023
Summary: In survival analysis, the relationship between a survival time and a covariate is conveniently modeled with the proportional hazards regression model. This model usually assumes that the covariate has a log-linear effect on the hazard function. We consider the proportional hazards regression model with a nonparametric risk effect. We discuss estimation of the risk function and its derivatives in two cases: when the baseline hazard function is parametrized and when it is not parametrized.
In the case of a parametric baseline hazard function, inference is based on a local version of the likelihood function, while in the case of a nonparametric baseline hazard, we use a local version of the partial likelihood. This results in maximum local likelihood estimators and maximum local partial likelihood estimators, respectively. We establish the asymptotic normality of the estimators. It turns out that both methods have the same asymptotic bias and variance in a common situation, even though the local likelihood method uses information about the baseline hazard function.

MSC:
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
60G44 Martingales with continuous parameter
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[1] AITKIN, M. and CLAy TON, D. G. 1980. The fitting of exponential, Weibull and extreme value distributions to complex censored survival data using GLIM. J. Roy. Statist. Soc. Ser. C 29 156 163. Z. · Zbl 0437.62092
[2] ANDERSEN, P. K., BORGAN, Ø., GILL, R. D. and KEIDING, N. 1993. Statistical Models Based on Counting Processes. Springer, New York. Z. · Zbl 0769.62061
[3] 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
[4] BRESLOW, N. E. 1972. Comment on “Regression and life tables,” by D. R. Cox. J. Roy. Statist. Soc. Ser. B 34 216 217. Z.
[5] BRESLOW, N. E. 1974. Covariance analysis of censored survival data. Biometrics 30 89 99. Z. Z.
[6] COX, D. R. 1972. Regression models and life-tables with discussion. J. Roy. Statist. Soc. Ser. B 4 187 220. Z. JSTOR: · Zbl 0243.62041
[7] COX, D. R. 1975. Partial likelihood. Biometrika 62 269 276. Z. JSTOR: · Zbl 0312.62002
[8] FAN, J. and GIJBELS, I. 1996. Local Poly nomial Modelling and Its Applications. Chapman and Hall, London. Z. · Zbl 0873.62037
[9] FAN, J., HARDLE, W. and MAMMEN, E. 1995. Direct estimation of additive and linear compo\"nents for high dimensional data. Inst. Statist. Mimeo Series 2339. Univ. North Carolina, Chapel Hill. Z.
[10] FLEMING, T. R. and HARRINGTON, D. P. 1991. Counting Processes and Survival Analy sis. Wiley, New York. Z.
[11] GENTLEMAN, R. and CROWLEY, J. 1991. Local full likelihood estimation for the proportional hazards model. Biometrics 47 1283 1296. Z. JSTOR:
[12] HASTIE, T. and TIBSHIRANI, R. 1990a. Generalized Additive Models. Chapman and Hall, London. Z. · Zbl 0747.62061
[13] HASTIE, T. and TIBSHIRANI, R. 1990b. Exploring the nature of covariate effects in the proportional hazards model. Biometrics 46 1005 1016. Z.
[14] HJORT, N. L. 1996. Dy namic likelihood hazard rate estimation. Biometrika. To appear. Z.
[15] KOOPERBERG, C., STONE, C. J. and TRUONG, Y. 1995a. Hazard regression. J. Amer. Statist. Assoc. 90 78 94. Z. JSTOR: · Zbl 0818.62097
[16] KOOPERBERG, C., STONE, C. J. and TRUONG, Y. 1995b. The L rate of convergence for hazard 2 regression. Scand. J. Statist. 22 143 157. Z. · Zbl 0839.62050
[17] LEHMANN, E. L. 1983. Theory of Point Estimation. Wadsworth & Brooks Cole, Pacific Grove, CA. Z. · Zbl 0522.62020
[18] LI, G. and DOSS, H. 1995. An approach to nonparametric regression for life history data using local linear fitting. Ann. Statist. 23 787 823. Z. · Zbl 0852.62037
[19] LINTON, O. and NIELSEN, J. P. 1995. A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93 100. Z. JSTOR: · Zbl 0823.62036
[20] MARRON, J. S. and NOLAN, D. 1988. Canonical kernels for density estimation. Statist. Probab. Lett. 7 195 199. Z. · Zbl 0662.62035
[21] MARRON, J. S. and PADGETT, W. J. 1987. Asy mptotically optimal bandwidth selection from randomly right-censored samples. Ann. Statist. 15 1520 1535. Z. · Zbl 0657.62038
[22] MULLER, H. G. and WANG, J. L. 1990. Analy zing changes in hazard functions: an alternative to \" change-point models. Biometrika 77 610 625. Z.
[23] MULLER, H. G. and WANG, J. L. 1994. Hazard rate estimation under random censoring with \" varying kernels and bandwidths. Biometrics 50 61 76. Z. O’SULLIVAN, F. 1988. Nonparametric estimation of relative risk using splines and crossvalidation. SIAM J. Sci. Statist. Comput. 9 531 542. Z.
[24] POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045
[25] STONE, C. J. 1994. The use of poly nomial splines and their tensor products in multivariate Z. function estimation with discussion. Ann. Statist. 22 118 184. · Zbl 0827.62038
[26] STUTE, W. and WANG, J. L. 1993. A strong law under random censorship. Ann. Statist. 21 1591 1607. Z. · Zbl 0785.60020
[27] TIBSHIRANI, R. and HASTIE, T. 1987. Local likelihood estimation. J. Amer. Statist. Assoc. 82 559 567. Z. JSTOR: · Zbl 0626.62041
[28] WONG, W. H. 1986. Theory of partial likelihood. Ann. Statist. 14 88 123. · Zbl 0603.62032
[29] CHAPEL HILL, NORTH CAROLINA 27599-3260
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