Let be the number of events experienced by a subject over the time interval . Let be the increment of the counting process N over . The rate function is given by
where is an unknown continuous function. If the are time independent, the rate model is reduced to the mean model
The partial likelihood for n independent triplets , of counting, atrisk, and covariate processes is the same as that of the multiplicative hazards model. However, a robust sandwich estimate is used for the covariance matrix of the parameter estimator instead of the modelbased estimate.
Let be the kth event time of the ith subject. Let be the censoring time of the ith subject. The atrisk indicator and the failure indicator are, respectively,
Denote
Let be the maximum likelihood estimate of , and let be the observed information matrix. The robust sandwich covariance matrix estimate is given by
where






For a given realization of the covariates , the Nelson estimator is used to predict the mean function
with standard error estimate given by
where









Since the cumulative mean function is always nonnegative, the log transform is used to compute confidence intervals. The % pointwise confidence limits for are
where is the upper percentage point of the standard normal distribution.