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Time-dependent coefficients in a Cox-type regression model. (English) Zbl 0754.62069
A model relating an output counting process $$N$$ to an input covariate process $$X$$ which is often used in survival analysis, is the Cox regression model. This model stipulates that the stochastic intensity of $$N$$ is $$\lambda(t,X)=e^{\beta X(t)}\lambda_ 0(t)$$. Here, the regression coefficient, $$\beta$$, is an unknown scalar and $$\lambda_ 0$$ is an unspecified deterministic function. Since $$\beta$$ is constant in time, the above model implies that the effect of $$X$$ on the underlying stochastic intensity $$\lambda_ 0$$ (and hence on $$N$$) is time invariant. This is not always the case, particularly in survival analysis applications. In the survival analysis setting several authors have considered a time-varying regression coefficient.
The method presented here, which also allows $$\beta$$ to be infinite dimensional, utilizes the method of sieves [U. Grenander, Abstract inference. (1981; Zbl 0505.62069)], and in particular, a very simple sieve, the histogram sieve. This choice of a sieve retains the simplicity of analysis present in methods involving only a finite-dimensional parameterization of the regression coefficient $$\beta$$. In addition, the estimation method presented is applicable not only in the survival analysis context where $$N$$ can have at most one jump, but also in the more general context where $$N$$ is allowed multiple jumps.
Section 1 contains a description of the statistical model with a list of assumptions made in the following theorems. Weak consistency (with a rate of convergence) is proved in Section 2. Next in Section 3, both a functional central limit theorem for the integrated regression coefficient and a consistent estimator of the asymptotic variance process are given. Section 4 provides conditions under which the theorems in Sections 2 and 3 hold in the independent and identically distributed case. In Section 5, extensions to the multivariate model and to the R. L. Prentice and S. G. Self model [Ann. Stat. 11, 804-813 (1983; Zbl 0526.62017)] are discussed. The last section contains the technical details.

##### MSC:
 62M09 Non-Markovian processes: estimation 62E20 Asymptotic distribution theory in statistics 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P10 Applications of statistics to biology and medical sciences; meta analysis 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F17 Functional limit theorems; invariance principles
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