On kernel mode estimation under RLT and WOD model

Abstract

Let $(X_N)_{N\geq 1}$ denote a sequence of real random variables and let $\vartheta$ be the mode of the random variable of interest $X$. In this paper, we study the kernel mode estimator (say) $\vartheta_n$ when the data are widely orthant dependent (WOD) and subject to Random Left Truncation (RLT) mechanism. We establish the uniform consistency rate of the density estimator (say) $f_n$ of the underlying density $f$ as well as the almost sure convergence rate of $\vartheta_n$. The performance of the estimators are illustrated via some simulation studies and applied on a real dataset of car brake pads.

Figures (8)

• Figure 1: Kernel density estimator : n=50 and TR $\approx$$10 \%$, $30 \%$, $50 \%$
• Figure 2: Kernel density estimator : n=100 and TR $\approx$$10 \%$, $30 \%$, $50 \%$
• Figure 3: Kernel density estimator : n=500 and TR $\approx$$10 \%$, $30 \%$, $50 \%$
• Figure 4: Kernel mode estimator : n=50 and TR $\approx$$10 \%$, $30 \%$, $50 \%$
• Figure 5: Kernel mode estimator : n=100 and TR $\approx$$10 \%$, $30 \%$, $50 \%$

Theorems & Definitions (10)

• Remark 1
• Proposition 1
• Theorem 2
• Remark 2
• Theorem 3
• Remark 3
• Lemma 1
• Remark 4
• Lemma 2
• Lemma 3