Nonparametric inference for rates with censored survival data. (English) Zbl 0598.62050

Yandell, Brian S.

Ann. Stat. 11, 1119-1135 (1983).

This paper develops asymptotic confidence bands and hypotheses tests for the hazard rate function h(x) under the random censorship model which assumes that the lifetimes and censoring times for the n sampled individuals constitute independent and identically distributed random pairs. Attention is focused on estimators of the form \[ h_ n(x)=\int K_ b(x,y)dH_ n(y), \] where \(H_ n(\cdot)\) is the Nelson-Aalen empirical cumulative hazard rate, \(K_ b\) is a kernel of the form w[(x- y)/b]/b with w(\(\cdot)\) a density, and the band-width \(b\to 0\) and nb\(\to \infty\) as \(n\to \infty\). Strong approximation results are derived for the pivot process \[ W_ n(x)=[nb/V_ h(x)]^{1/2}[h_ n(x)-E h_ n(x)], \] where \(V_ h(x)=\lim nb Var(h_ n(x))\). The limiting process is then inverted to derive simultaneous asymptotic confidence bands for h(x). Further applications of the strong approximation include a goodness-of-fit test for hypotheses of the form \(h(x)=h_ 0(x,\theta)\), and a two-sample test for the equality of the hazard rates. The effects of censoring on the bias, variance and maximum absolute deviation are investigated by simulations with exponential survival and censoring distributions. An application is illustrated with a data set from a survival experiment with serial sacrifice.

MSC:

62G15	Nonparametric tolerance and confidence regions
62N05	Reliability and life testing
62G10	Nonparametric hypothesis testing
62G05	Nonparametric estimation

Keywords:

life testing; censored survival data; asymptotic confidence bands; hazard rate function; random censorship model; censoring; estimators; Nelson- Aalen empirical cumulative hazard rate; Strong approximation results; pivot process; goodness-of-fit test; two-sample test for the equality of the hazard rates