On the estimation of the quantile density function by orthogonal series

Abstract

The classical estimator of a quantile density function by orthogonal series depends on the empirical distribution function estimator H_n. The fact that H_n is a step function even when the underlying cumulative distribution function $H(.)$ is continuous, has called for the need (in certain areas of application like estimating the quantile density function $\psi (.)$) for smooth estimators of H_n. The present work has two goals. The first one is to introduce a new technique for estimating $\psi (.)$ by orthogonal series for any orthonormal system in $L_2[0,1]$, a smooth nonparametric estimators of $\psi (.)$ and $H(.)$ are proposed. Asymptotic properties of the proposed estimators are studied. The second is to introduce a new method for selection of a smoothing parameter. A simulation study is done to compare the performances of the new approach with the (Chesneau et al 2016) one, when comparing mean integrated square error of the two estimators.

Keywords:

• Nonparametric quantile function density
• cumulative distribution and density
• trigonometric series
• orthogonal series

Mathematics Subject Classification:

• 62G32
• 62G05
• 62G30
• 62G32

Acknowledgments

The authors would like to thank the anonymous referees for their careful readings and helpful comments that led to a considerable improvement of the paper.

Appendix

Proof of theorem 4.

$$\left|{\widehat{H}}_{d_n}\right(x)-\mathbb{E}\left({\widehat{H}}_{d_n}\left(x\right)\right)|=\left|\sum _{k=0}^{d_n}{\widehat{A}}_k{e}_k\right(x)-\sum _{k=0}^{d_n}{A}_k{e}_k(x\left)\right|$$

Then, $$\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-\mathbb{E}({\widehat{H}}_{d_n}\left(x\right)\left)\right|\le \underset{x\in [0,1]}{\sup }\left|e_k\right(x\left)\right|\sum _{k=0}^{d_n}|{\widehat{A}}_k-A_k|$$

In addition, $$\mathbb{E}\left[{\left(\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-\mathbb{E}\left({\widehat{H}}_{d_n}\left(x\right)\right)|\right)}^2\right]\le {\left(\underset{x\in [0,1]}{\sup }\left|e_k\right(x\left)\right|\right)}^2\mathbb{E}{\left(\sum _{k=0}^{d_n}|{\widehat{A}}_k-A_k|\right)}^2$$

According to the Theorem 1 and the Corollary 1 hence, $$\underset{n\to \infty }{\lim }\mathbb{E}\left[{\left(\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-\mathbb{E}\left({\widehat{H}}_{d_n}\left(x\right)\right)|\right)}^2\right]=0$$

Consequently, $$\underset{n\to \infty }{\lim }\left\{\mathbb{E}{\left[{\left(\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-\mathbb{E}\left({\widehat{H}}_{d_n}\left(x\right)\right)|\right)}^2\right]}^{\frac{1}{2}}\right\}=0$$

However, $$\underset{n\to \infty }{\lim }{\widehat{H}}_{d_n}\left(x\right)=H\left(x\right)$$

It follow that: $$\underset{n\to \infty }{\lim }\mathbb{E}\left[{\left(\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-H(x\left)\right|\right)}^2\right]=0$$

Then, we deduce that $$\underset{n\to \infty }{\lim }P\left[\underset{x\in [0,1]}{\sup }\left|{\widehat{H}}_{d_n}\right(x)-H(x\left)\right|\hspace{0.17em}<\hspace{0.17em}ϵ\right]=1,\forall ϵ\hspace{0.17em}>\hspace{0.17em}0$$

Table 1. Optimal smoothing parameters and optimal MISE for estimator with cosine basis.