Smoothed jackknife empirical likelihood for the difference of two quantiles

Abstract

In this paper, we propose a smoothed estimating equation for the difference of quantiles with two samples. Using the jackknife pseudo-sample technique for the estimating equation, we propose the jackknife empirical likelihood (JEL) ratio and establish the Wilk’s theorem. Due to avoiding estimating link variables, the simulation studies demonstrate that JEL method has computational efficiency compared with traditional normal approximation method. We carry out a simulation study in terms of coverage probability and average length of the proposed confidence intervals. A real data set is used to illustrate the JEL procedure.

Appendix: Proofs of Theorems

Proof of Theorem 1

We can decompose $\Pi _{m ,n}\{p,\theta (p)\}$ as

$$\begin{aligned} \Pi _{m ,n}\{p,\theta (p)\}&= \frac{1}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} -p-\Pi _{m }\{p,\theta (p)\}+\Pi _{m }\{p,\theta (p)\} , \end{aligned}$$

(2)

where

$$\begin{aligned} \Pi _{m }\{p,\theta (p)\}=\frac{1}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{2}\{X_{j}-\theta (p)\}}{h}\right\} -p. \end{aligned}$$

(3)

The Eq. (3) is simplified as follows

$$\begin{aligned} \Pi _{m }\{p,\theta (p)\}&= \frac{1}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{2}\{X_{j}-\theta (p)\}}{h}\right\} -p \nonumber \\&= \int _{-\infty }^{\infty } K\left\{ \frac{p-{F}_{2}\{x-\theta (p)\}}{h}\right\} \mathrm {d}F_{m,1}(x)-p\nonumber \\&= K\left\{ \frac{p-{F}_{2}\{x-\theta (p)\}}{h} \right\} F_{m,1}(x)|^{\infty }_{-\infty } \nonumber \\&\quad -\int _{-\infty }^{\infty } F_{m,1}(x)\mathrm {d}K\left\{ \frac{p-{F}_{2}\{x-\theta (p)\}}{h}\right\} -p\nonumber \\&= \frac{1}{h}\int _{-\infty }^{\infty } F_{m,1}(x)w\left\{ \frac{p-{F}_{2}\{x-\theta (p)\}}{h}\right\} \mathrm {d}{F}_{2} \{x-\theta (p)\}-p \nonumber \\&= \int _{-1}^{1} F_{m,1}\{F_2^{-1}(p+uh)+\theta (p)\}w\left( u\right) \mathrm {d}u-p \nonumber \\&=\int _{-1}^{1} [ F_{m,1}\{F_2^{-1}(p+uh)+\theta (p)\}- F_{1}\{F_2^{-1}(p+uh)+\theta (p)\} \nonumber \\&\quad + F_{1}\{F_2^{-1}(p+uh)+\theta (p)\} -F_{1}\{F_2^{-1}(p)+\theta (p)\}]w\left( u\right) \mathrm {d}u\nonumber \\&= o_p(1). \end{aligned}$$

(4)

The above equation is obtained by the Glivenko–Cantelli Theorem of $F_1$ and the bounded derivative of $D\{\theta (p), p\}=F_1\{F_2^{-1}(p)+\theta (p)\}$. By Eqs. (10) and (11) in Gong et al. (2010), we can extend their result in our case, i.e., $\Pi _{m}\{p,\theta (p)\}-\Pi _{m,n}\{p,\theta (p)\}=o_p(1)$. Thus, considering Eqs. (3) and (4), we finish the proof about

$$\begin{aligned} \Pi _{m ,n}\{p,\theta (p)\} \mathop { \rightarrow }\limits ^\mathcal{P} 0. \end{aligned}$$

(5)

$\square$

Proof of Theorem 2:

One has that

$$\begin{aligned} \sqrt{m+n}\Pi _{m,n}\{p,\theta (p)\}&=\frac{\sqrt{m+n}}{\sqrt{m}}\sqrt{m}[\Pi _{m} \{p,\theta (p)\}] \nonumber \\&\quad + \frac{\sqrt{m+n}}{\sqrt{n}}\sqrt{n}[\Pi _{m ,n}\{p,\theta (p)\}-\Pi _{m }\{p,\theta (p)\}]. \end{aligned}$$

(6)

For the first term of (6), one has

$$\begin{aligned}&\sqrt{m}[\Pi _{m }\{p,\theta (p)\}]\nonumber \\&\quad = \int _{-1}^{1}\sqrt{m} [ F_{m,1}\{F_2^{-1}(p+uh)+\theta (p)\}- F_{1}\{F_2^{-1}(p+uh)+\theta (p)\}]w\left( u\right) \mathrm {d}u \nonumber \\&\qquad + \sqrt{m}\int _{-1}^{1} F_{1}\{F_2^{-1}(p+uh)+\theta (p)\} -F_{1}\{F_2^{-1}(p)+\theta (p)\}]w\left( u\right) \mathrm {d}u\nonumber \\&\quad =\int _{-1}^{1}W_{F_{1}}\{F_2^{-1}(p+uh)+\theta (p)\}w\left( u\right) \mathrm {d}u +\sqrt{m}\int _{-1}^{1}D^{'}\{\theta (p),p\}uhw\left( u\right) \mathrm {d}u\nonumber \\&\qquad +O_p(\sqrt{m}h^2) = I+II+O_p(\sqrt{m}h^2), \end{aligned}$$

(7)

where $W_{F_1}(t)=\sqrt{m}\{F_{m,1}(t)-F_{1}(t)\}$. Because of the symmetric property of kernel function, the second term of (7) is equal to zero.

By arguments in pp. 266 and 269 in van der Vaart (2000), the weak convergence of $F_{m,1}(x)$ and $F_{n,2}(x)$ is true.

$$\begin{aligned} \sqrt{m}\{F_{m,1}(x)-F_1(x)\}\Longrightarrow B_{1}(F_1(x)),\quad \sqrt{n}\{F_{n,2}(x)-F_2(x)\}\Longrightarrow B_{2}(F_2(x)), \end{aligned}$$

where $B_{1}(\cdot )$ and $B_{2}(\cdot )$ are two independent Brownian bridge on [0, 1]. Thus, $B_{1}(F_1(x))$ and $B_{2}(F_2(x))$ are independent. Due to the Donsker theorem and similar proofs for equation 9 in Gong et al. (2010), $I\overset{\mathfrak {D}}{\longrightarrow }B_1(F_1\{F_2^{-1}(p)+\theta (p)\})$. Using $F_1^{-1}(p)=F_2^{-1}(p)+\theta (p)$, it is clear that

$$\begin{aligned} \sqrt{m}[\Pi _{m}\{p,\theta (p)\}]\overset{\mathfrak {D}}{\longrightarrow }B_1(p). \end{aligned}$$

(8)

For the second term of Eq. (6), under condition C.1, we adopt the procedure similar to that in Gong et al. (2010),

$$\begin{aligned}&\sqrt{n}[\Pi _{m ,n}\{p,\theta (p)\}-\Pi _{m}\{p,\theta (p)\}]\nonumber \\&\quad =-\int _{-\infty }^{\infty }W_{F_2}(x)w\left\{ \frac{p-{F}_{2}\{x-\theta (p)\}}{h}\right\} \mathrm {d}F_{1}(x)+O_p(n^{-1/2}h^{-1})\nonumber \\&\quad =\int _{-1}^{1}W_{F_2}\{F_2^{-1}(p)\}w(u)D^{'}\{p,\theta (p)\} \mathrm {d}u+O_p(n^{-1/2}h^{-1})\nonumber \\&\quad \overset{\mathfrak {D}}{\longrightarrow }B_2(F_2\{F_2^{-1}(p)\})D^{'}\{p,\theta (p)\},\nonumber \\&\quad =B_2(p)D^{'}\{p,\theta (p)\}, \end{aligned}$$

(9)

where $W_{F_2}(t)=\sqrt{n}\{F_{n,2}(t)-F_{2}(t)\}$. Combining (7), (8), (9) and the independence of $B_1(F_1(x))$ and $B_2(F_2(x))$, one has that

$$\begin{aligned} \sqrt{m+n}\Pi _{m,n}\{p,\theta (p)\}\overset{\mathfrak {D}}{\longrightarrow } N(0,\sigma ^{2}(p)). \end{aligned}$$

(10)

$\square$

Before we prove Theorem 3, we need to obtain the asymptotic normality of jackknife estimator and the consistency of jackknife variance estimator. Those asymptotic properties are given in Lemmas 1 and 2.

Lemma 1

Suppose conditions C.1–C.4 hold. We have

$$\begin{aligned} \sqrt{m+n}\left\{ \frac{1}{m+n}\sum ^{m+n}_{i=1}\hat{V}_{i}\{p, \theta (p)\}\right\} \mathop { \rightarrow }\limits ^\mathcal{D} N\{0, \sigma ^2(p)\}, \end{aligned}$$

where $\sigma ^{2}(p)$ is defined in Theorem 2.

Proof of Lemma 1:

First, we introduce some properties of $F_{n,2,-i}$ as follows:

$$\begin{aligned} F_{n,2,-i}(X_{j})-F_{n,2}(X_{j})=\frac{1}{n-1}\{F_{n,2}(X_{j})-I(Y_{ i} \le X_{j})\}=O_{p}\left( \frac{1}{n-1}\right) , i=1,\ldots ,n \end{aligned}$$

(11)

and

$$\begin{aligned} \sum ^{n}_{i=1}\{F_{n,2,-i}(X_{j})-F_{n,2}(X_{j})\}=0, \end{aligned}$$

because

$$\begin{aligned}&F_{n,2,-i}(X_{j})-F_{n,2}(X_{j})\nonumber \\&\quad = \frac{1}{n-1}\sum ^{n}_{k=1,k\ne i}I(Y_{k} \le X_{j})-\frac{1}{n} \sum ^{n}_{k=1}I(Y_{k} \le X_{j})\nonumber \\&\quad = \frac{1}{n-1}\left\{ \sum ^{n}_{k=1,k\ne i}I(Y_{k} \le X_{j})-\sum ^{n}_{k=1}I(Y_{ k} \le X_{j})\right\} +\left( \frac{1}{n-1}-\frac{1}{n} \right) \sum ^{n}_{k=1}I(Y_{k} \le X_{j})\nonumber \\&\quad = \frac{1}{n-1}\{F_{n,2}(X_{j})-I(Y_{ i} \le X_{j})\}. \end{aligned}$$

(12)

For the pseudo-sample, based on equation (16) in Gong et al. (2010) and Eqs. (11) and (12), one has that

$$\begin{aligned}&\left\{ \frac{1}{m+n}\sum ^{m+n}_{i=1}\hat{V}_{i}\{p, \theta (p)\}\right\} \nonumber \\&\quad = \frac{1}{m+n} \sum ^{m}_{i=1} \left\{ \frac{m+n}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right\} \nonumber \\&\qquad -\frac{m+n-1}{(m+n)(m-1)}\sum ^{m}_{i=1} \left\{ \sum ^{m}_{j=1, j\ne i} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right\} -p \nonumber \\&\qquad + \sum ^{m+n}_{i=m+1} \left\{ \frac{1}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right. \nonumber \\&\left. \qquad -\frac{m+n-1}{(m+n)m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2,m-i}\{X_{j}-\theta (p)\}}{h}\right\} \right\} \nonumber \\&\quad =\frac{1}{m+n} \left[ \sum ^{m}_{j=1} K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} +\frac{n}{m}\sum ^{m}_{j=1} K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right] \nonumber \\&\qquad + \frac{m+n-1}{(m+n)m} \sum ^{m+n}_{i=m+1}\sum ^{m}_{j=1} \left[ K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right. \nonumber \\&\left. \qquad - K\left\{ \frac{p-{F}_{n,2,m-i}\{X_{j}-\theta (p)\}}{h}\right\} \right] -p\nonumber \\&\quad =\frac{1}{m}\sum ^{m}_{j=1} K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} -p+O_p\left\{ \frac{mn}{(m+n)(n-1)^2h}\right\} . \end{aligned}$$

(13)

Using (10) and (13), it is clear that

$$\begin{aligned} \sqrt{m+n}\left\{ \frac{1}{m+n}\sum ^{m+n}_{i=1}\hat{V}_{i}\{p, \theta (p)\}\right\}&=\sqrt{m+n}\Pi _{m ,n}\{p,\theta (p)\}+o_p(1)\nonumber \\&\overset{\mathfrak {D}}{\longrightarrow }N(0,\sigma ^{2}(p)). \end{aligned}$$

(14)

$\square$

In order to obtain the Wilks’ theorem for the JEL procedure, we need to check the consistency of jackknife pseudo-sample variance in addition to Eq. (14). Define the pseudo-sample variance

$$\begin{aligned} v^2_{m,n}\{p,\theta (p)\}=\frac{1}{m+n}\sum ^{m+n}_{i=1}\left\{ \hat{V}_{i} \{p, \theta (p)\}-\frac{1}{m+n}\sum ^{m+n}_{i=1}\hat{V}_{i}\{p, \theta (p)\}\right\} ^{2}. \end{aligned}$$

Lemma 2

Under the conditions C.1–C.4, one has that

$$\begin{aligned} v^2_{m ,n}\{p,\theta (p)\} \mathop { \rightarrow }\limits ^\mathcal{P} \sigma ^2(p). \end{aligned}$$

Proof of Lemma 2:

For $1\le i\le m$,

$$\begin{aligned}&\hat{V}_{i}\{p, \theta (p)\}\\&\quad =\frac{m+n}{m} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} -\frac{m+n-1}{m-1} \sum ^{m}_{j=1, j\ne i} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} -p \\&\quad = \frac{m+n-1}{m-1} K \left\{ \frac{p-{F}_{n,2}(X_{i}-\theta (p))}{h}\right\} - \frac{n}{m(m-1)}\sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} -p \end{aligned}$$

and

$$\begin{aligned}&\hat{V}_{i}^2\{p, \theta (p)\}\\&=\left[ \frac{m+n-1}{m-1} K \left\{ \frac{p-{F}_{n,2}(X_{i}-\theta (p))}{h}\right\} \right] ^2+\left[ \frac{n}{m(m-1)}\sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right] ^2\\&\quad -\frac{2(m+n-1)n}{m(m-1)^2} K \left\{ \frac{p-{F}_{n,2}(X_{i}-\theta (p))}{h}\right\} \sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} +p^2\\&\quad - 2p\left[ \frac{m+n-1}{m-1} K \left\{ \frac{p-{F}_{n,2}(X_{i}-\theta (p))}{h}\right\} - \frac{n}{m(m-1)}\sum ^{m}_{j=1} K \left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right] . \end{aligned}$$

By the similar argument in Gong et al. (2010), one has

$$\begin{aligned} \frac{1}{m+n}\sum ^{m}_{j=1}\hat{V}_{i}^2\{p, \theta (p)\}\overset{\mathcal {P}}{\longrightarrow }\frac{r+1}{r}p(1-p). \end{aligned}$$

(15)

For $m+1\le i\le m+n$, one has that

$$\begin{aligned}&\hat{V}_{i}\{p, \theta (p)\} \\&\quad =\frac{m+n-1}{m} \sum ^{m}_{j=1} \left[ K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} - K\left\{ \frac{p-{F}_{n,2,m-i}\{X_{j}-\theta (p)\}}{h}\right\} \right] \\&\qquad +\frac{1}{m}\sum ^{m}_{j=1}K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} - p, \end{aligned}$$

and

$$\begin{aligned}&\hat{V}^2_{i}\{p, \theta (p)\} =\left\{ \frac{m+n-1}{m}\right\} ^2 \left[ \sum ^{m}_{j=1} K\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \right. \nonumber \\&\left. \qquad - K\left\{ \frac{p-{F}_{n,2,m-i}\{X_{j}-\theta (p)\}}{h}\right\} \right] ^2+o_p(1) =\left\{ \frac{m+n-1}{m}\right\} ^2 \nonumber \\&\qquad \times \left[ \sum ^{m}_{j=1} w\left\{ \frac{p-{F}_{n,2}\{X_{j}-\theta (p)\}}{h}\right\} \frac{{F}_{n,2}\{X_{j}-\theta (p)\}-{F}_{n,2,m-i}\{X_{j}-\theta (p)\}}{h}\right] ^2\\&\qquad + o_p(1). \end{aligned}$$

Under condition C.1, we follow the argument which is similar to Gong et al. (2010),

$$\begin{aligned}&\frac{1}{m+n}\sum ^{m+n}_{j=m+1}\hat{V}^2_{i}\{p, \theta (p)\} = \frac{m+n}{nh^2}\sum ^n_{i=1}\sum ^m_{j=1}\sum ^m_{l=1} \{F_{n,2}(X_{j}\nonumber \\&\quad -\theta (p))F_{n,2}(X_{l}-\theta (p))-F_{n,2}\{X_{j}-\theta (p)\}I(Y_{i} \le X_{l}-\theta (p))\\&\quad -F_{n,2}(X_{l}-\theta (p))I(Y_{i} \le X_{j}-\theta (p))+I(Y_{i} \le X_{j}-\theta (p)) I(Y_{i} \le X_{l}-\theta (p))\}\\&\quad w\left( \frac{p-F_{n,2}\{X_{j}-\theta (p)\}}{h}\right) w\left( \frac{p-F_{n,2} (X_{l}-\theta (p))}{h}\right) +o_p(1) \end{aligned}$$

$$\begin{aligned}&\quad =\frac{m+n}{nh^2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\{F_{n,2}(x_{1}\wedge x_{2}-\theta (p))-F_{n,2}(x_{1}-\theta (p))F_{n,2}(x_{2}-\theta (p))\}\nonumber \\&\qquad w\left( \frac{p-F_{n,2}(x_{1}-\theta (p))}{h}\right) w \left( \frac{p-F_{n,2}(x_{2}-\theta (p))}{h}\right) \mathrm {d} F_{m,1}(x_1)\mathrm {d} F_{m,1}(x_2)+o_p(1)\nonumber \\&\quad =\frac{m+n}{nh^2}\int _{-1}^{1}\int _{-1}^{1}\{F_{2}\{F_{2}^{-1}(p-u_1h)\wedge F_{2}^{-1}(p-u_2h)\}\nonumber \\&\qquad -F_{2}\{F_{2}^{-1}(p-u_1h)\}F_{2}\{F_{2}^{-1}(p-u_2h)\}\}\nonumber \\&\qquad w\left( u_1\right) w\left( u_2\right) \mathrm {d} F_{1}\{F_{2}^{-1}(p-u_1h)+\theta (p)\}\mathrm {d} F_{1}\{F_{2}^{-1}(p-u_2h)+\theta (p)\}+o_p(1)\nonumber \\&\quad =\frac{m+n}{n}\int _{-1}^{1}\int _{-1}^{1}p(1-p)\{D^{'}\{p,\theta (p)\}\}^2w(u_1)w(u_2)\mathrm {d}u_1\mathrm {d}u_2\nonumber \\&\quad =\frac{m+n}{n}p(1-p)\{D^{'}\{p,\theta (p)\}\}^2+o_p(1). \end{aligned}$$

(16)

Thus, based on Eqs. (15) and (16),

$$\begin{aligned}&\frac{1}{m+n}\sum ^{m+n}_{j=1}\hat{V}^2_{i}\{p, \theta (p)\}\overset{\mathcal {P}}{\longrightarrow }\sigma ^2(p). \end{aligned}$$

From Eqs. (5) and (13),

$$\begin{aligned} v^2_{m ,n}\{p,\theta (p)\} \mathop { \rightarrow }\limits ^\mathcal{P} \sigma ^2(p). \end{aligned}$$

$\square$

Proof of Theorem 3:

Combining Lemmas 1 and 2, we can easily prove Theorem 3 by the standard arguments in Owen (1990). The details of the proof are omitted. $\square$