On the performance of weighted bootstrapped kernel deconvolution density estimators

Abstract

We propose a weighted bootstrap approach that can improve on current methods to approximate the finite sample distribution of normalized maximal deviations of kernel deconvolution density estimators in the case of ordinary smooth errors. Using results from the approximation theory for weighted bootstrap empirical processes, we establish an unconditional weak limit theorem for the corresponding weighted bootstrap statistics. Because the proposed method uses weights that are not necessarily confined to be uniform (as in Efron’s original bootstrap), it provides the practitioner with additional flexibility for choosing the weights. As an immediate consequence of our results, one can construct uniform confidence bands, or perform goodness-of-fit tests, for the underlying density. We have also carried out some numerical examples which show that, depending on the bootstrap weights chosen, the proposed method has the potential to perform better than the current procedures in the literature.

Appendix

1.1 Derivation of (25)

The derivation of (25) is similar to, and easier than, that of (31) given below.

1.2 Derivation of (31)

Since $\big |\widehat{g}_{nn}(t)-\widehat{g}_n(t)\big |\ge \left( \big |\widehat{g}_{nn}(t)-\widehat{g}_n(t)\big |/\,\big |\widehat{g}_{nn}(t)\big |\right) \cdot \inf _{t\in [0,1]} |\widehat{g}_{nn}(t)|$, we find that for all $n>0$

$$\begin{aligned} \sup _{t\in [0,1]}\big |\widehat{g}_{nn}(t)-g_n(t)\big |\ge & {} \sup _{t\in [0,1]}\left| \frac{\widehat{g}_{nn}(t)-g_n(t)}{\widehat{g}_{nn}(t)}\right| \cdot \inf _{t\in [0,1]} |\widehat{g}_{nn}(t)| \end{aligned}$$

(32)

Now, observe that for all $n>0$

$$\begin{aligned}&-\sup _{t\in [0,1]}\big |\widehat{g}_{nn}(t)-\widehat{g}_n(t)\big | \le \inf _{t\in [0,1]} \big (|\widehat{g}_{nn}(t)|- |\widehat{g}_n(t)|\big ) \le \inf _{t\in [0,1]} \big |\widehat{g}_{nn}(t)- \widehat{g}_n(t)\big | \\&\quad \le \sup _{t\in [0,1]}\big |\widehat{g}_{nn}(t)-\widehat{g}_n(t)\big |. \end{aligned}$$

But, as $n\rightarrow \infty$, both the far left and the far right sides of the above chain of inequalities converge to zero (by (30)) and, hence, so does the middle term, i.e., $\inf _{t\in [0,1]} \big (|\widehat{g}_{nn}(t)|-|\widehat{g}_n(t)|\big )\rightarrow _p 0$. This together with part (a) of Assumption (A3) and the fact that $\sup _{t\in [0,1]}|\widehat{g}_n(t)-g(t)|\rightarrow _p0$ for the kernel density estimator $\widehat{g}_n$ defined in (17), imply that

$$\begin{aligned} \lim _{n\rightarrow \infty }\inf _{t\in [0,1]} |\widehat{g}_{nn}(t)|&= \lim _{n\rightarrow \infty }\inf _{t\in [0,1]} \big \{|\widehat{g}_{nn}(t)|- |\widehat{g}_n(t)|+|\widehat{g}_n(t)|-g(t)+ g(t)\big \}\\&\quad \ge \lim _{n\rightarrow \infty }\big \{\inf _{t\in [0,1]} \big (|\widehat{g}_{nn}(t)|- |\widehat{g}_n(t)|\big )+\inf _{t\in [0,1]}\big (|\widehat{g}_n(t)|-g(t)\big )\\&\qquad +\inf _{t\in [0,1]} g(t)\big \}\\&\quad =\,_p 0 + 0 + \inf _{t\in [0,1]} g(t) >0, \end{aligned}$$

where we have used the simple inequality that for bounded functions $\psi _1, \psi _2: \mathcal {C}\rightarrow \mathbb {R}$, where $\mathcal {C}\subset \mathbb {R}$, one has $\inf _{t\in \mathcal {C}} \{\psi _1(t)+\psi _2(t)\}\ge \inf _{t\in \mathcal {C}} \psi _1(t) +\inf _{t\in \mathcal {C}} \psi _2(t)$. Next let $\{a_n\}$ be any sequence of positive numbers converging to zero and observe that if we multiply both sides of the inequality (32) by $a_n\sqrt{n\lambda _n/\log n}$, then upon taking the limit as $n\rightarrow \infty$, we find (in view of (30)) that $a_n\sqrt{n\lambda _n/\log n}\;\sup _{t\in [0,1]} \big |\big (\widehat{g}_{nn}(t)- \widehat{g}_n(t)\big )/\,\widehat{g}_{nn}(t)\big |\rightarrow _p 0$. Now, (31) follows because the sequence $\{a_n\}$ can be chosen to converge arbitrarily slowly. $\square$