Time-dependent coefficients in a Cox-type regression model.

*(English)*Zbl 0754.62069A 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.

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 |

##### Keywords:

point process; martingales; output counting process; input covariate process; survival analysis; Cox regression model; stochastic intensity; time-varying regression coefficient; method of sieves; histogram sieve; multiple jumps; weak consistency; rate of convergence; functional central limit theorem; consistent estimator of the asymptotic variance process
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\textit{S. A. Murphy} and \textit{P. K. Sen}, Stochastic Processes Appl. 39, No. 1, 153--180 (1991; Zbl 0754.62069)

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