Given the moment condition dMi(t) for estimating the regression coefficients β in Cox’s proportional hazards model (Eq. (1)), we have learned from this study the important roles of zi and¯z(t)in the optimal m.e.f. U?T(β)(Eqs. (6) and (8)) respectively: (1) zi comes from the derivatives ∂ [dMi(t)]/∂β, and thus it is expected to be orthogonal todMi(t) given the unbiasedness of the m.e.f.; and (2) centering zi by ¯z(t) helps us get rid of the nuisance parameterλ0(t). Since the statistical methodologies adopted in this study including the GMM estimation method, theory of e.f.’s, adaptive estimation, and martingales have not yet exerted their full powers in our case, the finding of this study is not only very interesting in its own rights, but it provides us with an opportunity to develop GLMs-type regression models locally
(at each time t) for more general stochastic processes and to apply some powerful GMM-related estimating techniques such as the instrumental variables (IV) method for nonlinear equations to deal with several known statistical modeling problems in analysis of survival or time-to-event data.
In particular, Bowden and Turkington (1984, esp., Sec. 1.2, pp. 10-16), Greene (2000, Sec. 9.5 and Subsecs. 10.3.2, 11.5.5, 13.7.3, and 16.5.2, pp. 370-387, 430-438, 483-488, 550-552, and 680-690), and Newey (1990) among others gave nice reviews and discussions on the IV estimation method and its applications in econometric models. In the linear equation settings, the IV method has proved its power in solving the estimation problems occurring when some of the covariates in the equation arecorrelatedwith equation’s error. Thus, it can be used to deal with four kinds of statistical modeling problems: (1) the error-in-variable (or measurement error) problem, (2) the self-selection problem, (3) the simultaneous-equations bias, and (4) the time series problem (Bowden and Turkington 1984, pp. 3-10). Then, based on the result of this study, we are specifically interested in developing IV estimators for the regression coefficients β in Cox’s proportional hazards model to deal with the measurement error problem and simultaneous-equations bias respectively. It is our understanding that when these problems arise in Cox’s proportional hazards model, some of the covariates zi would notbe orthogonal todMi(t). And, for this purpose, our IV estimation method for GLMs (Hu, et. al. 2002) may be applied to Cox’s proportional hazards model locally (at each timet).
5 ACKNOWLEDGEMENTS
This paper is based on the first part of the second author’s dissertation under the first
author’s advice. And, this research was financially supported by the National Science Council of the Republic of China (NSC 91-2118-M-002-004).
6 APPENDIX
We shall give a sketch of the rationale behind the proposed asymmetric orthogonal expected information approach to dealing with the nuisance parameter λ0(t) for an adaptive estimation of the interested parametersβ. Yet, this general approach can possibly be applied to many different settings. Without loss of generality, we assume thatλ0(t)can be parameterized by a finite number of parameters for simplicity. Let the estimating functions for β and λ0(t) be denoted as U?β(β, λ0(t)) and Uλ0(β, λ0(t)) respectively, of which both are functions of
(β, λ0(t)). And, given a random sample of size n, we assume that both U?β(β, λ0(t)) and
Uλ0(β, λ0(t)) are unbiased, the solutions(eβ?, eλ0(t)) to these two estimating equations exist, and they are consistent estimates of (β, λ0(t)).
Thekey conditionrequired for adaptively estimating β (see Eq. (4) in the text) is that at the true values of β,
E h
I?βλ0 i
= − E
∂U?β(β, λ0(t))
∂λ0(t)
= 0.
Then, symbolically, the expected values of the minus mixed derivatives of the joint estimating functions U?β(β, λ0(t)) and Uλ0(β, λ0(t)) with respect to the parameters(β, λ0(t))are
E
I?ββ I?βλ
0
Iλ0β Iλ0λ0
=
i?ββ 0 iλ0β iλ0λ0
which may be non-symmetric. And, with a zero on the upper right-hand corner of the above
partitioned matrix, it is straightforward to show that
Then, under the regularity conditions,
n1/2
Thus, by picking up the first p elements in the vector on the left-hand side,
n1/2(eβ? − β•) ∼=
According to the
multivariate central limit theorem
,n−1/2U?β(β•, λ•0(t)) −−−→D N
Therefore,
n1/2(eβ?− β•) −−−→D N 0, lim
n→∞
I?ββ(β•, λ•0(t)) n
−1!
which is exactly the same asymptotic distribution ofβe? as if λ0(t) were known.
7 REFERENCES
Andersen, P. K., Borgan, Ø., Gill, R. D. & Keiding, N. (1993).Statistical Models Based on Counting Processes. New York, NY: Springer-Verlag.
Andersen, P. K. & Gill, R. D. (1982). Cox’s regression model for counting process: A large sample study. Ann. Statist.
10
, 1100-20.Begun, J. M., Hall, W. J., Huang, W. & Wellner, J. A. (1983). Information and asymptotic efficiency in parametric-nonparametric models. Ann. Statist.
11
, 432-53.Bowden, R. J. & Turkington, D. A. (1984). Instrumental Variables. Cambridge: Cambridge University Press.
Chang, I. -S. & Hsiung, C. A. (1990). Finite sample optimality of maximum partial likelihood estimation in Cox’s model for counting process. J. Statist. Plan. Inf.
25
, 35-42.Chang, I. -S. & Hsiung, C. A. (1991). Applications of estimating function theory to repli-cates of generalized proportional hazards models. In: V. P. Godambe (Ed), Estimating Functions, Oxford: Clarendon Press, pp. 23-33.
Cox, D. R. (1972). Regression models and life tables (with discussion).J. Roy. Statist. Soc., Ser. B
34
, 187-220.Cox, D. R. (1975). Partial likelihood.Biometrika
62
, 269-76.Desmond, A. F. (1991). Quasi-likelihood, stochastic processes, and optimal estimating func-tions. In: V. P. Godambe (Ed), Estimating Functions, Oxford: Clarendon Press, pp.
133-46.
Fleming, T. R. & Harrington, D. D. (1991).Counting Processes and Survival Analysis. New York, NY: John Wiley & Sons.
Gill, R. D. (1984). Understanding Cox’s regression model: A martingale approach.J. Amer.
Statist. Assoc.
79
, 441-7.Godambe, V. P. (1960). An optimum property of regular maximum likelihood estimation.
Ann. Math. Statist.
31
, 1208-12.Godambe, V. P. (1985). The foundations of finite sample estimation in stochastic processes.
Biometrika
72
, 419-28.Godambe, V. P. & Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Intern.
Statist. Rev.
55
, 231-44.Godambe, V. P. & Kale, B. K. (1991). Estimating functions: An overview. In: V. P. Godambe (Ed), Estimating Functions, Oxford: Clarendon Press, pp. 3-20.
Greene, W. H. (2000).Econometric Analysis, 4th ed. Upper Saddle River, NJ: Prentice-Hall.
Greenwood, P. E. & Wefelmeyer, W. (1991). On optimal estimating functions for partially specified counting process models. In: V. P. Godambe (Ed), Estimating Functions, Oxford: Clarendon Press, pp. 147-60.
Heyde, C. C. (1997).Quasi-Likelihood and Its Application: A General Approach to Optimal Parameter Estimation. New York, NY: Springer-Verlag.
Hu, F. -C., Lai, S. -H., Tsai, T. -L. & Shau, W. -Y. (2002). The method of instrumental vari-ables for generalized linear models. Technical report. Division of Biostatistics, Graduate Institute of Epidemiology, College of Public Health, National Taiwan University, Taipei, Taiwan, R.O.C..
Khuri, A. I. (1993).Advanced Calculus with Applications in Statistics. New York, NY: John Wiley & Sons.
McCullagh, P. (1991). Quasi-likelihood and estimating functions. In: D. V. Hinkley, N. Reid
& E. J. Snell (Eds), Statistical Theory and Modelling: In Honour of Sir David Cox, FRS, London: Chapman and Hall, pp. 265-86.
McCullagh, P. & Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. London: Chap-man & Hall.
Moore, D. F. (1986). Asymptotic properties of moment estimators for overdispersed counts and proportions. Biometrika
73
, 583-8.Nelder, J. A. & Wedderburn, R. W. M. (1972). Generalized linear models. J. Roy. Statist.
Soc., Ser. A
135
, 370-84.Newey, W. K. (1990). Efficient instrumental variable estimation of nonlinear models. Econo-metrica
58
, 809-38.Pagan, A. & Ullah, A. (1999). Nonparametric Econometrics. New York, NY: Cambridge University Press.
Prentice, R. L. & Self, S. (1983). Asymptotic distribution theory for Cox-type regression models with general relative risk form. Ann. Statist.
11
, 804-13.Small, C. G. & McLeish, D. L. (1994).Hilbert Space Methods in Probability and Statistical Inference. New York, NY: John Wiley & Sons.
Thavanewswaran, A. & Thompson, M. E. (1986). Optimal estimation for semimartingales.
J. Appl. Probab.
23
, 409-17.Tsiatis, A. A. (1981). A large sample study of Cox’s regression model. Ann. Statist.
9
, 93-108.Wedderburn, R. W. M. (1974). Quasilikelihood functions, generalized linear models and the Gauss-Newton method.Biometrika
61
, 439-47.Williams, D. A. (1982). Extra-binomial variation in logistic linear models.Appl. Statist.