Strong Consistency Rate for the Kernel Mode Estimator Under Strong Mixing Hypothesis and Left Truncation

Abstract

Let Y N N ≥1 be a sequence of copies of a random variable of interest Y . In this article, we study the kernel estimator, sayˆ n , of the mode of Y when Y is subject to the random left truncation. While based on n n ≤ N actual observations fulfilling the well-known -mixing condition, we establish the strong consistency with a rate of the proposed estimatorˆ n .

A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data

“…Using the triangular inequality it can be seen that Under Assumptions A3, A5, H1-H4, Theorem 4.1 of [16] we obtain From (4.45) and (4.52) we have Using (4.53) and (4.52) in (4.51) proves the corollary. □…”

“…A3(iii) is specially needed for a Taylor expansion. H1-H4 are needed to use Theorem 4.1 of [16] in Theorem 4 here.…”

“…According to Lemma 9 for any we can write and From Lemma 5.2 of [16] we have and from [22] we have So we can write Now by choosing and using Theorem 1 we get the result  □…”

A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data

“…The following Lemma is Lemma 4.2 in Ould-Saïd and Tatachak [22], in which they state a rate of convergence for µ n under α-mixing hypothesis, which is interesting in itself, similar to that established in the iid case by He and Yang [13]. Proof.…”

“…The bandwidth h n =: h satisfies: and (D3) are needed for covariance calculus and take similar forms to those used under mixing. Hypothesis (H2) is used in Ould-Saïd and Tatachak [22] and is needed to establish Lemma 5.1 and Lemma 5.4. Assumptions (M) concern the mixing processes structure which are standard in such situation.…”