Kernel-Smoothed Hazard Estimate

Kernel-smoothed estimators of the hazard function are based on the Nelson-Aalen estimator and its variance . Consider the jumps of and at the event times as follows:

where =0.

The kernel-smoothed estimator of is a weighted average of over event times that are within a bandwidth distance of . The weights are controlled by the choice of kernel function, , defined on the interval [–1,1]. The choices of are as follows:

• uniform kernel

• Epanechnikov kernel

• biweight kernel

The kernel-smoothed hazard rate estimator is defined for all time points on . For time points for which , the kernel-smoothed estimated of based on the kernel is given by

The variance of is estimated by

For , the symmetric kernels are replaced by the corresponding asymmetric kernels of Gasser and Müller (1979). Let . The modified kernels are as follows:

• uniform kernel

• Epanechnikov kernel

• biweight kernel

For , let . The asymmetric kernels for are used with replaced by .

Using the log transform on the smoothed hazard rate, the 100(1–)% pointwise confidence interval for the smoothed hazard rate is given by

where is the 100(1–)th percentile of the standard normal distribution.

Optimal Bandwidth

The following mean integrated squared error (MISE) over the range and is used as a measure of the global performance of the kernel function estimator

The last term is independent of the choice of the kernel and bandwidth and can be ignored when you are looking for the best value of . The first integral can be approximated by using the trapezoid rule by evaluating at a grid of points . You can specify , and by using the options MISEMIN=, MISEMAX=, and MISENUM=, respectively, of the HAZARD plot. The second integral can be estimated by Ramlau-Hansen (1983a, 1983b) cross-validation estimate

Therefore, for a fixed kernel, the optimal bandwidth is the quantity that minimizes

The minimization is carried out by the golden section search algorithm.