On the strong uniform consistency of a conditional mode estimator for randomly left truncated time series

Let (YN) N≥1 denote a sequence of random variables of interest and (XN) N≥1 be a sequence of Rd-valued covariates. Let denote the conditional mode of Y given X= x. In the present paper, we study a kernel conditional mode estimator (say) of the conditional mode of a randomly left truncated variable Y. Given a sample (Xi, Yi), 1 ≤ i ≤ n (n ≤ N), of truncated replicates of (X, Y), which fulfill the well-known α mixing condition, the goal is to establish the strong uniform consistency of the proposed estimator as well as the convergence rate.
Key words: Kernel conditional mode estimator, Lynden-Bell estimator, random left-truncation model, strong mixing condition, uniform almost sure convergence.
2000 MSC: 62G05, 62G07, 62G20