Asymptotic Properties of Maximum Partial Likelihood Estimators When The Relative Risk Possesses First Order Derivative

Knots in a free-knot spline model are treated as parameters and can be threshold values such as changepoints. Motivated by quadratic splines which only possess continuous first order derivatives, this article investigates the asymptotic properties of a general semiparametric multiplicative hazard model when the relative risk is expressed as a first order continuously differentiable parametric function. It is shown that the logarithm of the partial likelihood function of the model is locally concave under suitable conditions. Using the convexity lemma [Andersen and Gill [1], Pollard [2]], it is proved that the maximum partial likelihood estimators of parameters uniquely exist in a neighborhood of the true values of parameter and are consistent and asymptotically normal. The developed theory is applied to derive the asymptotic normality of the maximum partial likelihood estimators of parameters in a free-knot quadratic spline model.