A large sample study of rank estimation for censored regression data. (English) Zbl 0773.62048

Ying, Zhiliang

Ann. Stat. 21, No. 1, 76-99 (1993).

Summary: Large sample approximations are developed to establish asymptotic linearity of the commonly used linear rank estimating functions, defined as stochastic integrals of counting processes over the whole line, for censored regression data. These approximations lead to asymptotic normality of the resulting rank estimators defined as solutions of the linear rank estimating equations.
A second kind of approximations is also developed to show that the estimating functions can be uniformly approximated by certain more manageable nonrandom functions, resulting in a simple condition that guarantees consistency of the rank estimators. This condition is verified for the two-sample problem, thereby extending earlier results of T. A. Louis [Biometrika 68, 381-390 (1981; Zbl 0469.62035)] and L. J. Wei and M. H. Gail [J. Am. Stat. Assoc. 78, 382-388 (1983; Zbl 0586.62049)], as well as in the case when the underlying error distribution has increasing failure rate, which includes most parametric regression models in survival analysis. Techniques to handle the delicate tail fluctuations are provided and discussed in detail.

MSC:

62J05	Linear regression; mixed models
62F12	Asymptotic properties of parametric estimators
62M99	Inference from stochastic processes
60F05	Central limit and other weak theorems
62E20	Asymptotic distribution theory in statistics
62N05	Reliability and life testing
62G05	Nonparametric estimation

Keywords:

accelerated life model; censored regression; log-rank statistic; asymptotic normality; large sample approximations; asymptotic linearity; linear rank estimating functions; stochastic integrals of counting processes; censored regression data; rank estimators; linear rank estimating equations; nonrandom functions; consistency; two-sample problem; increasing failure rate; survival analysis; tail fluctuations