Statistical Generalized Derivative Applied to the Profile Likelihood Estimation in a Mixture of Semiparametric Models.

There is a difficulty in finding an estimate of the standard error (SE) of the profile likelihood estimator in the joint model of longitudinal and survival data. The difficulty is on the differentiation of an implicit function that appear in the profile likelihood estimation. We solve the difficulty by introducing the "statistical generalized derivative". The derivative is used to show the asymptotic normality of the estimator with the SE expressed in terms of the profile likelihood score function.