Adaptive Functional Local Polynomial Estimation of the Conditional Hazard Function Under Missing-at-Random Responses

Abstract

We propose an adaptive functional local polynomial estimator of the conditional hazard function of a scalar response, observed only partially, given a functional covariate valued in a semi-metric space. The estimator extends the functional local linear method of Kenouza, Mechab and Benaissa to a polynomial order that grows with the sample size through the adaptive rule p_n = ⌊log n⌋. As in the local linear case, the conditional distribution and the conditional density are estimated first, the hazard estimator being their ratio. Under missing-at-random (MAR) responses and the same regularity setting as the local linear analysis (small-ball probability, Hölder-type continuity, standard kernel and bandwidth conditions, together with a non-degeneracy condition for the growing design), we establish the almost complete convergence with rate of the three estimators, following the same four-step proof strategy: identification of the conditional law on complete cases, control of the bias by the Hölder exponents, an exponential inequality for the stochastic term, and control of the denominator. The growing order is shown to be a smoothness-adaptive device: it automatically attains the local-polynomial bias rate permitted by the (unknown) functional regularity, while the slow logarithmic growth keeps the local design matrix stable. A Monte Carlo study illustrates the finite-sample behaviour under several missingness levels.