The Bahadur Representation for Empirical and Smooth Quantile Estimators Under Association

Abstract

In this paper, the Bahadur representation of the empirical and Bernstein polynomials estimators of the quantile function based on associated sequences are investigated. The rate of approximation depends on the rate of decay in covariances, and it converges to the optimal rate observed under independence when the covariances quickly approach zero. As an application, we establish a Berry-Esseen bound with the rate $O(n^{-1/3})$ assuming polynomial decay of covariances. All these results are established under fairly general conditions on the underlying distributions. Additionally, we perform Monte Carlo simulations to evaluate the finite sample performance of the suggested estimators, utilizing an associated and non-mixing model.

Appendix

This appendix contains supplementary lemmas that are essential parts of providing a more comprehensive understanding of the paper.

Lemma 3.5

(Birkel (1988b), Lemma 3.1) Let A and B be finite sets and let $(X_j)_{j\in A\cup B}$ be associated random variables. Then for all real-valued partially differentiable functions $g(\cdot )$, $h(\cdot )$ with bounded partial derivatives, there holds

$$\begin{aligned} \Big | \textrm{Cov}\big ( g\big ( (X_i)_{i\in A}\big ), h\big ( (X_j)_{j\in B}\big ) \big ) \Big |\le \sum _{i\in A}\sum _{j\in B} \left\| \frac{ \partial g}{\partial x_i} \right\| _{\infty }\left\| \frac{ \partial h}{\partial x_j}\right\| _{\infty } \textrm{Cov}(X_i,Y_j). \end{aligned}$$

Lemma 3.7

(Shao and Yu (1996), Lemma 5.1) Let X and Y be associated random variables with a common uniform distribution over [0, 1]. Then for any $0\le s < t \le 1$,

$$\begin{aligned} \Big |\textrm{Cov}\big (\mathbbm {1}_{\{s<X\le t\}}, \mathbbm {1}_{\{s<Y\le t\}}\big )\Big | \le 4 (t-s)^{\frac{1}{3}} \big (\textrm{Cov}(X,Y)\big )^{\frac{1}{3}}. \end{aligned}$$

Lemma 7.1

(Shao and Yu (1996), (5.27)) Let $q>2$. Let $(U_i)_{i\ge 1}$ be a stationary associated sequence of uniform [0, 1] random variables and let

$$\begin{aligned} Z_i=\mathbbm {1}_{\{s<U_i\le t\}}-(t-s),~ \text{ for } ~ 0\le s<t\le 1. \end{aligned}$$

If $2 n^{\frac{-q+1+\eta }{q+2}}<t-s$ and $\textrm{Cov}(U_1,U_n)=O(n^{-b})$ for some $b>q-1$, then, for any $\eta >0$, there exists some positive constant $K_{\eta }$ for which

$$\begin{aligned} \textbf{E}\left| \sum _{i=1}^{n} Z_i\right| ^{q}\le K_{\eta } \left\{ n^{\frac{q(3+\eta )}{q+2}}+\left( n\sum _{i=1}^{n} \big |\textbf{E Z}_1 Z_i\big |\right) ^{\frac{q}{2}}\right\} . \end{aligned}$$

Lemma 7.2

(Newman and Wright (1981), (12)) Suppose that $X_1,\ldots , X_m$ are associated, mean zero, finite variance, random variables. Then for any real number $\lambda$

$$\begin{aligned} \textbf{P}\Big (\max \big \{|S_1|,\ldots ,|S_m|\big \}\ge \lambda s_m\Big )\le 2 \textbf{P}\Big (|S_m|\ge (\lambda - \sqrt{2}) s_m\Big ), \end{aligned}$$

where

$$\begin{aligned} S_m=\sum _{i=1}^{m} X_i \end{aligned}$$

and $s_m^2=\textbf{E S}_m^2$.

Lemma 7.3

(Birkel (1988a), Theorem 2) Let $(X_i)_{i\in \mathbb {N}}$ be a sequence of random variables satisfying $\textbf{E X}_i=0$ and $|X_i|\le C <\infty$ for $i\in \mathbb {N}$. Assume for some $r>2$

$$\begin{aligned}\ \sup _{k\in \mathbb {N} } \sum _{i: |i-k|\ge n}\textrm{Cov}(X_{i}, X_{k})= O\left( n^{-\frac{r-2}{2}}\right) \quad n\in \mathbb {N}. \end{aligned}$$

Then there is a constant B not depending on n, such that for all $n \ge 1$,

$$\begin{aligned} \sup _{m\in \mathbb {N} }\textbf{E}\left| \sum _{i=m+1}^{m +n } X_i \right| ^r \le B n^{\frac{r}{2}}. \end{aligned}$$

Lemma 7.4

(Móricz (1983), Corollary 1) Let $\alpha >1$, $\gamma \ge 1$ and $d\ge 1$. Let $\{\xi _i, \, i\in \mathbb {N}^{d}\}$ be real random fields having finite moments of order $\gamma$. Suppose that there exists a nonnegative and superadditive function $g(R_{b,p})$ of the rectangle

$$\begin{aligned} R_{b,p}=\bigg \{(i_1,\ldots ,i_d)\in \mathbb {N}^{d}: b_j+1\le i_j \le b_j+p_j, \, j=1,\ldots , d\bigg \}, \end{aligned}$$

where $b_j\in \mathbb {N}$ and $1\le p_j\le m_j$, $j=1,\ldots , d$, such that for every $R_{b,p}$ we have

$$\begin{aligned} \textbf{E}\left| \sum _{i\in R_{b,p}} \xi _i\right| ^{\gamma }\le \big (g(R_{b,p}))^{\alpha }. \end{aligned}$$

Then for every $R_{b,p}$ we have

$$\begin{aligned} \textbf{E}\left( \max _{1\le p_1 \le m_1}\ldots \max _{1\le p_d\le m_d}\left| \sum _{i_1=b_1+1}^{b_1+p_1}\ldots \sum _{i_d=b_d+1}^{b_d+p_d}\xi _{i_1,\ldots ,i_d}\right| \right) ^{\gamma }\le C(\alpha ,\gamma ,d) \big (g(R_{b,m})\big )^{\alpha }, \end{aligned}$$

where

$$\begin{aligned} C(\alpha ,\gamma ,d)=\left( \frac{5}{2}\right) ^d \left( 1-2^{(1-\alpha )\gamma }\right) ^{-d\gamma }. \end{aligned}$$

Lemma 8.1

(Louhichi (2001), Theorem 1) Let $(X_i)_{i\in \mathbb {N}}$ be a stationary sequence of centered and bounded associated random variables with a second finite moment such that

• i)
  $$\begin{aligned} \exists n_0 \in \mathbb {N}^{*}\quad \text {such that}\quad \inf _{n\ge n_0,\, 1\le k<n} \frac{\displaystyle \textrm{Var}\left( \sum _{i=1}^{n} X_i\right) -\textrm{Var}\left( \sum _{i=1}^{k} X_i\right) }{\displaystyle n-k}>0; \end{aligned}$$
• ii)
  $$\begin{aligned}\ \sum _{i=1}^{\infty } i^{\delta }\textrm{Cov}(X_1, X_{1+i})<\infty , \quad 0<\delta \le 1. \end{aligned}$$

Then, for any $n\ge n_0$, we have

$$\begin{aligned} \sup _{x\in \mathbb {R}} \left| \textbf{P}\left( \frac{\displaystyle \sum _{i=1}^{n} X_i}{\sqrt{\textrm{Var}\left( \displaystyle \sum _{i=1}^{n} X_i\right) }}\le x\right) -\Phi (x)\right| = O( n^{-\delta /3}). \end{aligned}$$