Nonparametric inference for rates with censored survival data
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.
Nelson- Aalen empirical cumulative hazard rate
two-sample test for the equality of the hazard rates
scientific article; zbMATH DE number 3965206