New non-parametric inferences for low-income proportions

Abstract

Low-income proportion is an important index in describing the inequality of an income distribution. It has been widely used by governments in measuring social stability around the world. Established inferential methods for this index are based on the empirical estimator of the index. It may have poor finite sample performances when the real income data are skewed or has outliers. In this paper, based on a smooth estimator for the low-income proportion, we propose a smoothed jackknife empirical likelihood approach for inferences of the low-income proportion. Wilks theorem is obtained for the proposed jackknife empirical likelihood ratio statistic. Various confidence intervals based on the smooth estimator are constructed. Extensive simulation studies are conducted to compare the finite sample performances of the proposed intervals with some existing intervals. Finally, the proposed methods are illustrated by a public income dataset of the professors in University System of Georgia.

Appendix: Proof of theorems

The Proof of Theorem 2.1

We have the following decomposition

$$\begin{aligned} \sqrt{n} \{{{\hat{T}}}_n(\alpha , \beta ) - \theta _{\alpha \beta } \}= & {} \sqrt{n} \left[ \frac{1}{n} \sum _{i=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - \frac{1}{n} \sum _{i=1}^n K\left( \frac{\alpha \xi _{\beta } - X_i}{h}\right) \right] \nonumber \\&+\sqrt{n} \left[ \frac{1}{n} \sum _{i=1}^n K\left( \frac{\alpha \xi _{\beta } - X_i}{h}\right) - F(\alpha \xi _\beta )\right] \equiv I_1 + I_2. \end{aligned}$$

(5)

$I_1$ from (5) can be written as

$$\begin{aligned} I_1= & {} \int _{-\infty }^{\infty } \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) - K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) \right] \mathrm{d} \left[ \sqrt{n}(F_n(x)-F(x))\right] \nonumber \\&+ \sqrt{n}\int _{-\infty }^{\infty } \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) - K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) \right] \mathrm{d}F(x) \equiv I_{11} + I_{12}. \end{aligned}$$

(6)

Then using Taylor expansion and the Bahadur’s representation for sample quantile (Bahadur 1966)

$$\begin{aligned} {{\hat{\xi }}}_{\beta } - \xi _{\beta } = \frac{\beta - \frac{1}{n} \sum _{i=1}^n I(X_i \le \xi _{\beta })}{f(\xi _{\beta })} + o_p(n^{-\frac{1}{2}}), \end{aligned}$$

$I_{12}$ from (6) can be written as

$$\begin{aligned} I_{12}= & {} \sqrt{n}\int _{-\infty }^{\infty } \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) - K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) \right] \mathrm{d}F(x)\nonumber \\= & {} \sqrt{n}\int _{-\infty }^{\infty } \omega \left( \frac{\alpha \xi _{\beta } - x}{h}\right) \frac{\alpha {{\hat{\xi }}}_{\beta } - \alpha \xi _{\beta }}{h}\, \mathrm{d}F(x) + o_p(1) \nonumber \\= & {} - \sqrt{n}\int _{-\infty }^{\infty } \omega \left( \frac{\alpha \xi _{\beta } - x}{h}\right) \frac{\alpha }{h} \frac{\frac{1}{n} \sum _{i=1}^n I(X_i \le \xi _{\beta }) - \beta }{f(\xi _{\beta })}\,\mathrm{d}F(x) + o_p(1) \nonumber \\= & {} -\frac{\alpha f(\alpha \xi _\beta )U_n(\beta )}{f(\xi _{\beta })} + o_p(1), \end{aligned}$$

(7)

where $U_n(\beta ) = \sqrt{n}[\frac{1}{n} \sum _{i=1}^nI(X_i \le \xi _{\beta }) - \beta ] = \sqrt{n}[\frac{1}{n} \sum _{i=1}^nI(F(X_i) \le {\beta }) - \beta ].$

Since $\sqrt{n}[F_n(x)-F(x)]\rightarrow B(x)$, which is a Gaussian process, $\sqrt{n} ({{\hat{\xi }}}_{\beta } - \xi _{\beta })=O_p(1)$, and $\sqrt{n} h \rightarrow \infty$, we get $I_{11} = o_p(1)$. Therefore, $I_1=-\frac{\alpha f(\alpha \xi _\beta )U_n(\beta )}{f(\xi _{\beta })} + o_p(1)$.

Next, let us consider $I_2$ from (5). We are going to prove

$$\begin{aligned} EK \left( \frac{\alpha \xi _\beta - X}{h}\right) \mathop {\longrightarrow }\limits ^{{}} \theta _{\alpha \beta },\quad \text{ and }\quad EK^2 \left( \frac{\alpha \xi _\beta - X}{h}\right) \mathop {\longrightarrow }\limits ^{{}} \theta _{\alpha \beta },\qquad \hbox {as } h\rightarrow 0. \end{aligned}$$

Notice that

$$\begin{aligned}&\lim _{h \rightarrow 0} EK \left( \frac{\alpha \xi _\beta - X}{h}\right) = \lim _{h \rightarrow 0}\int _{-\infty }^{\infty } K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) f(x)\,\mathrm{d}x \nonumber \\&\quad = \lim _{h \rightarrow 0}\int _{-\infty }^{\infty } \int _{-\infty }^{\frac{\alpha \xi _{\beta } - x}{h}} \omega (y)\,\mathrm{d}y f(x)\,\mathrm{d}x = \int _{-\infty }^{\infty } \left[ \lim _{h \rightarrow 0} \int _{-\infty }^{\frac{\alpha \xi _{\beta } - x}{h}} \omega (y)\,\mathrm{d}y\right] f(x)\,\mathrm{d}x \nonumber \\&\quad = \int _{-\infty }^{\infty }\{0*I[\alpha \xi _{\beta } < x] \!+\! \int _{-\infty }^{0}\omega (y)\,\mathrm{d}y* I[\alpha \xi _{\beta } \!=\! x] +1*I[\alpha \xi _{\beta } > x] \}f(x)\,\mathrm{d}x\nonumber \\&\quad = \int _{-\infty }^{\infty } I[\alpha \xi _{\beta } > x ] f(x)\,\mathrm{d}x = \int _{-\infty }^{\infty } I[F(x)<F(\alpha \xi _\beta )]\,\mathrm{d}F(x) \nonumber \\&\quad = F(\alpha \xi _\beta ) = \theta _{\alpha \beta }. \end{aligned}$$

(8)

Similarly, we have

$$\begin{aligned}&\lim _{h \rightarrow 0} EK^2 \left( \frac{\alpha \xi _\beta - X}{h}\right) =\lim _{h \rightarrow 0}\int _{-\infty }^{\infty } K^2\left( \frac{\alpha \xi _{\beta } - x}{h}\right) f(x) \,\mathrm{d}x \nonumber \\&\quad = \int _{-\infty }^{\infty }\{0*I[\alpha \xi _{\beta } < x] \!+\! \int _{-\infty }^{0}\omega (y)\mathrm{d}y*I[\alpha \xi _{\beta } \!=\! x] \!+\!1*I[\alpha \xi _{\beta } > x] \}^2 f(x)\,\mathrm{d}x\nonumber \\&\quad = F(\alpha \xi _\beta ) = \theta _{\alpha \beta }. \end{aligned}$$

(9)

Let $W_i = K(\frac{\alpha \xi _{\beta } - X_i}{h})$. So $I_2$ from (5) can be rewritten as

$$\begin{aligned} I_2= & {} \sqrt{n} \left[ \frac{1}{n} \sum _{i=1}^n K\left( \frac{\alpha \xi _{\beta } - X_i}{h}\right) - EK\left( \frac{\alpha \xi _{\beta } - X}{h}\right) \right] + o_p(1) \nonumber \\= & {} \frac{1}{\sqrt{n}} \sum _{i=1}^n (W_i - EW_i) + o_p(1). \end{aligned}$$

(10)

Let $\{U_1, U_2,\ldots , U_n\}$ be an i.i.d. sample from U(0, 1) (uniform distribution on [0, 1]) and independent of $\{X_1, X_2,\ldots , X_n\}$. Since $U_i \mathop {=}\limits ^{{d}} F(X_i) \mathop {\sim }\limits ^{\mathrm{i.i.d.}} U(0,1)$ for any continuous distribution function F, then

(11)

where $\mathop {=}\limits ^{{d}}$ means that two statistics asymptotically have the same distribution. Therefore,

$$\begin{aligned} I_1 + I_2&\mathop {=}\limits ^{{d}}&\frac{1}{\sqrt{n}} \sum _{i=1}^n \left[ - \frac{\alpha f(\alpha \xi _\beta )}{f(\xi _{\beta })}\left( I(U_i \le {\beta }) - \beta \right) + (W_i - EW_i)\right] + o_p(1).\nonumber \\ \end{aligned}$$

(12)

Since

$$\begin{aligned}&\lim _{h \rightarrow 0} E[(I(U_i \le {\beta }) - \beta )(W_i - EW_i)] \\&\quad = \int _{-\infty }^{\infty }[I(F(x) \le {\beta }) - \beta ] \left[ \lim _{h \rightarrow 0} K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) - \lim _{h \rightarrow 0} EW_i \right] f(x)\,\mathrm{d}x \\&\quad = \int _{-\infty }^{\infty }[I(F(x) \le {\beta }) - \beta ][I[\alpha \xi _{\beta } > x ] - \theta _{\alpha \beta }]f(x)\,\mathrm{d}x \\&\quad = \theta _{\alpha \beta } (1-\beta ), \end{aligned}$$

$\hbox {Var}[I(U_i \le {\beta }) - \beta ]= \beta (1-\beta )$, $\hbox {Var}(W_i) = EK^2 (\frac{\alpha \xi _{\beta } - x}{h}) - [EK(\frac{\alpha \xi _{\beta } - x}{h})]^2 = \theta _{\alpha \beta }(1- \theta _{\alpha \beta }) +o(1)$, by the central limit theorem applicable to a triangular array setting (see Shao 2003), we get that

$$\begin{aligned} I_1 + I_2 \mathop {\longrightarrow }\limits ^{{d}} N(0, \sigma ^2_{\alpha \beta }), \end{aligned}$$

(13)

where $\sigma ^2_{\alpha \beta } = \frac{\alpha ^2 \beta (1-\beta ) f^2(\alpha \xi _ \beta )}{f^2(\xi _{\beta })} - 2\alpha (1-\beta )\theta _{\alpha \beta } \frac{f(\alpha \xi _{\beta })}{f(\xi _{\beta })} + \theta _{\alpha \beta } (1-\theta _{\alpha \beta })$. The proof of Theorem 2.1 is complete.

We need Lemmas 1 and 2 to prove Theorem 3.1.

Lemma 1

Under the conditions in Theorem 2.1, we have

$$\begin{aligned} \sqrt{n} \left\{ \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}(\alpha , \beta ) - \theta _{\alpha \beta }\right\} \mathop {\longrightarrow }\limits ^{{d}}N(0, \sigma ^2_{\alpha \beta }), \end{aligned}$$

(14)

where $\sigma ^2_{\alpha \beta }$ is defined in Theorem 2.1.

Proof

Note that $\frac{1}{n}\sum _{k=1}^n {{\hat{V}}}_{k}(\alpha , \beta )$ can be decomposed into

$$\begin{aligned} \frac{1}{n}\sum _{k=1}^n {{\hat{V}}}_{k}(\alpha , \beta )=\frac{n-1}{n}\sum _{k=1}^n[{{\hat{T}}}_n (\alpha , \beta ) - {{\hat{T}}_{n-1,k}}(\alpha , \beta )] + {{\hat{T}}}_n (\alpha , \beta ), \end{aligned}$$

(15)

while

$$\begin{aligned}&{{\hat{T}}}_n (\alpha , \beta ) - {{\hat{T}}_{n-1,k}} (\alpha , \beta ) \nonumber \\&\quad = \frac{1}{n}\sum _{i=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - \frac{1}{n-1 } \sum _{j \ne k}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) \nonumber \\&\quad = \frac{1}{n}\sum _{i=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - \frac{1}{n} \sum _{j \ne k}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) \nonumber \\&\qquad + \frac{1}{n} \sum _{j \ne k}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) - \frac{1}{n-1 } \sum _{j \ne k}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) . \end{aligned}$$

(16)

So

$$\begin{aligned}&\sum _{k=1}^n({{\hat{T}}}_n (\alpha , \beta ) - {{\hat{T}}_{n-1,k}}(\alpha , \beta ) ) \nonumber \\&\quad = \left\{ \frac{1}{n} \sum _{k=1}^n \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_i}{h}\right) \right] \right\} \nonumber \\&\qquad + \left\{ \frac{1}{n} \sum _{k=1}^n \sum _{j=1}^n \ \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) \right] \!-\! \frac{1}{n-1}\sum _{k=1}^n \sum _{j=1}^n \ \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_j}{h}\right) \right] \right. \nonumber \\&\qquad + \left. \frac{1}{n-1} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_k}{h}\right) \right\} \equiv I_1 + I_2. \end{aligned}$$

(17)

Using the Bahadur representation for sample quantile (Bahadur 1966), we get that

$$\begin{aligned} {{\hat{\xi }}}_{\beta ,-k} - {{\hat{\xi }}}_{\beta }= & {} ({{\hat{\xi }}}_{\beta ,-k} - \xi _{\beta } ) - ( {{\hat{\xi }}}_{\beta } - \xi _{\beta }) \nonumber \\= & {} \left[ \frac{\beta - \frac{1}{n-1} \sum _{j \ne k}^n I (X_j \le \xi _{\beta })}{f(\xi _{\beta })} \right] - \left[ \frac{\beta - \frac{1}{n} \sum _{i =1}^n I (X_i \le \xi _{\beta })}{f(\xi _{\beta })} \right] \nonumber \\&+o_p(n^{-1/2}) \nonumber \\= & {} \frac{ \frac{1}{n-1} [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )]}{f(\xi _{\beta })} + o_p(n^{-1/2}), \end{aligned}$$

(18)

and $(\frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h})^2 =O_p(\frac{1}{n^2h^2})$. Under conditions of Theorem 2.1, using Taylor expansion, we get that

$$\begin{aligned} I_1= & {} \frac{1}{n} \sum _{k=1}^n \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k}-X_i}{h}\right) \right] \nonumber \\= & {} \frac{1}{n}\sum _{k=1}^n \sum _{i=1}^n \left[ -\omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h}\right. \nonumber \\&\left. -\frac{1}{2}\omega ^{'} \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) \left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h} \right) ^2\right] +o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} \frac{1}{n} \sum _{i=1}^n \left\{ -\omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) \sum _{k=1}^n \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h}\right. \nonumber \\&\left. -\frac{1}{2}\sum _{k=1}^n\omega ^{'} \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) \left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h}\right) ^2\right\} +o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} - \frac{1}{2} \frac{1}{n} \sum _{i=1}^n \sum _{k=1}^n\omega ^{'} \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) \left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h}\right) ^2 + o_p(n^{-1/2}) +o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} O_p\left( \frac{1}{nh^2}\right) \int _{-\infty }^{\infty } \omega ^{'} \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) \mathrm{d}F_n(x) + o_p(n^{-1/2})+o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} O_p\left( \frac{1}{nh^2}\right) \int _{-\infty }^{\infty } \omega ^{'} \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) \mathrm{d}F(x) + o_p(n^{-1/2})+o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} O_p\left( \frac{1}{nh}\right) \int _{-\infty }^{\infty } \omega ^{'} (y) f(\alpha {{\hat{\xi }}}_{\beta }-yh) \mathrm{d}y + o_p(n^{-1/2})+o_p\left( \frac{1}{nh}\right) \nonumber \\= & {} O_p\left( \frac{1}{nh}\right) + o_p(n^{-1/2}). \end{aligned}$$

(19)

Meanwhile, $I_2$ from (17) can be written to

$$\begin{aligned} I_2= & {} \sum _{k=1}^n \left[ \left( \frac{1}{n}-\frac{1}{n-1}\right) \sum _{j=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_j}{h}\right) + \frac{1}{n-1}K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \right] \nonumber \\= & {} \frac{-1}{n(n-1)} \sum _{k=1}^n \sum _{j=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_j}{h} \right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_j}{h}\right) \right] \nonumber \\&+ \frac{-1}{n(n-1)} \sum _{k=1}^n \sum _{j=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_j}{h}\right) \nonumber \\&- \frac{-1}{n-1} \sum _{k=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \right] \nonumber \\&- \frac{-1}{n-1} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \nonumber \\= & {} O_p\left( \frac{1}{n(n-1)h}\right) - \frac{1}{n(n-1)} \sum _{k=1}^n \sum _{j=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_j}{h}\right) \nonumber \\&- O_p\left( \frac{1}{(n-1)^2h}\right) + \frac{1}{n-1} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \nonumber \\= & {} O_p\left( \frac{1}{n^2h}\right) . \end{aligned}$$

(20)

From (19) and (20), we get $I_1 + I_2 = O_p(\frac{1}{nh})+ o_p(n^{-1/2})$, which implies that

$$\begin{aligned} \frac{1}{n}\sum _{k=1}^n {{\hat{V}}}_k (\alpha , \beta )= & {} \frac{n-1}{n}\sum _{k=1}^n [{{\hat{T}}}_n (\alpha , \beta ) - {{\hat{T}}_{n-1,k}} (\alpha , \beta )] + {{\hat{T}}}_n (\alpha , \beta ) \nonumber \\= & {} {{\hat{T}}}_n (\alpha , \beta ) + O_p\left( \frac{1}{nh}\right) + o_p(n^{-1/2}). \end{aligned}$$

(21)

Therefore,

$$\begin{aligned} \sqrt{n} \left\{ \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_k (\alpha , \beta ) - \theta _{\alpha \beta } \right\}= & {} \sqrt{n} [{{\hat{T}}}_n (\alpha , \beta ) - \theta _{\alpha \beta }]\\&+ O_p\left( \frac{1}{\sqrt{n}h}\right) + o_p(1) \mathop {\longrightarrow }\limits ^{{d}} N(0, \sigma ^2_{\alpha \beta }). \end{aligned}$$

Lemma 2

Under the conditions in Theorem 2.1, we have that

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n \{ {{\hat{V}}}_{k}(\alpha , \beta ) - \theta _{\alpha \beta }\} ^2 \mathop {\longrightarrow }\limits ^{{p}} \sigma ^2_{\alpha \beta }. \end{aligned}$$

(22)

Proof

From Lemma 1, it follows that

$$\begin{aligned}&\frac{1}{n} \sum _{k=1}^n \{ {{\hat{V}}}_{k}(\alpha , \beta ) - \theta _{\alpha \beta }\} ^2 = \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}^2(\alpha , \beta )-2\theta _{\alpha \beta } \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}(\alpha , \beta ) + \frac{1}{n} \sum _{k=1}^n \theta ^2_{\alpha \beta } \nonumber \\&\quad \mathop {\longrightarrow }\limits ^{{p}} \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}^2(\alpha , \beta )-2\theta _{\alpha \beta } \theta _{\alpha \beta } + \frac{1}{n} n \theta ^2_{\alpha \beta } = \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}^2(\alpha , \beta ) - \theta ^2_{\alpha \beta }. \end{aligned}$$

(23)

By the definition of the jackknife pseudo-values for the low-income proportion, we have that

$$\begin{aligned} {{\hat{V}}}_{k}(\alpha , \beta )= & {} n {{\hat{T}}}_n(\alpha , \beta ) - (n-1) {{\hat{T}}}_{n-1,k} (\alpha , \beta ), \nonumber \\= & {} \sum _{i=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - \sum _{j \ne k}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_j}{h}\right) \nonumber \\= & {} \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_i}{h}\right) \right] + K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) ,\nonumber \\ \end{aligned}$$

(24)

and

$$\begin{aligned} {{\hat{V}}}_{k}^2(\alpha , \beta ) \!= & {} \! \left\{ \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } \!-\! X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} \!-\! X_i}{h}\right) \right] \right\} ^2 \!+\! K^2\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \nonumber \\&+ 2K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_i}{h}\right) \right] .\nonumber \\ \end{aligned}$$

(25)

Therefore,

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}^2(\alpha , \beta )= & {} \frac{1}{n} \sum _{k=1}^n \left\{ \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_i}{h}\right) \right] \right\} ^2 \nonumber \\&+ \frac{1}{n} \sum _{k=1}^n K^2\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \nonumber \\&+ \frac{2}{n} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \nonumber \\&\times \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_i}{h}\right) \right] \nonumber \\\equiv & {} J_1 + J_2 + J_3. \end{aligned}$$

(26)

Using Taylor expansion and the Bahadur representation for sample quantile, the first term $J_1$ in (26) can be written as

(27)

$J_2$ from (26) can be written as

$$\begin{aligned} J_2= & {} \frac{1}{n} \sum _{k=1}^n K^2\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \nonumber \\= & {} \frac{1}{n} \sum _{k=1}^n \left[ K^2\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) - K^2\left( \frac{\alpha \xi _{\beta } - X_k}{h}\right) \right] + \frac{1}{n} \sum _{k=1}^n K^2\left( \frac{\alpha \xi _{\beta } - X_k}{h}\right) \nonumber \\= & {} \frac{1}{n} \sum _{k=1}^n K^2\left( \frac{\alpha \xi _{\beta } - X_k}{h}\right) + o_p(1) \nonumber \\= & {} EK^2\left( \frac{\alpha \xi _{\beta } - x}{h}\right) + o_p(1) = \theta _{\alpha \beta } + o_p(1). \end{aligned}$$

(28)

The term $J_3$ from (26) can be written as

$$\begin{aligned} J_3= & {} \frac{2}{n} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \sum _{i=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_i}{h}\right) \right] \nonumber \\= & {} \frac{-2}{n} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta }}{h}\right) \sum _{i=1}^n \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_i}{h}\right) + o_p(1) \nonumber \\ \!= & {} \! \frac{-2\alpha }{nh} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} \!-\! X_k}{h}\right) ({{\hat{\xi }}}_{\beta ,-k} \!-\! {{\hat{\xi }}}_{\beta }) n \int _{-\infty }^{\infty } \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } \!-\! x}{h}\right) \mathrm{d}F(x) \!+\! o_p(1) \nonumber \\= & {} -2\alpha \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) ({{\hat{\xi }}}_{\beta ,-k} - {{\hat{\xi }}}_{\beta }) \int _{-\infty }^{\infty } \omega (z) f(\alpha \xi _{\beta } - zh)\,\mathrm{d}z + o_p(1) \nonumber \\= & {} -2\alpha \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) \frac{[ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )]}{(n-1) f(\xi _{\beta }) } f(\alpha \xi _\beta ) + o_p(1) \nonumber \\= & {} \frac{-2\alpha f(\alpha \xi _\beta ) }{f(\xi _{\beta })} \frac{1}{n-1} \sum _{k=1}^n \left[ K\left( \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - X_k}{h}\right) - K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \right. \nonumber \\&\left. + K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \right] [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )] + o_p(1) \nonumber \\= & {} \frac{-2\alpha f(\alpha \xi _\beta ) }{f(\xi _{\beta })} \left\{ \frac{1}{n-1} \sum _{k=1}^n \left[ - \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta } }{h} \right] \right. \nonumber \\&\times \left. [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )] \right. \nonumber \\&\left. + \frac{1}{n-1} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )] \right\} + o_p(1) \nonumber \\\equiv & {} \frac{-2\alpha f(\alpha \xi _\beta ) }{f(\xi _{\beta })} \{ M_1 + M_2\}+ o_p(1), \end{aligned}$$

(29)

and

$$\begin{aligned} M_1= & {} \frac{1}{n-1} \sum _{k=1}^n \left[ - \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \frac{\alpha {{\hat{\xi }}}_{\beta ,-k} - \alpha {{\hat{\xi }}}_{\beta } }{h} \right] [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )] \nonumber \\= & {} \frac{-\alpha }{(n-1)h} \sum _{k=1}^n \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) \frac{[ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )]^2}{(n-1) f(\xi _{\beta }) } + o_p(1) \nonumber \\= & {} \frac{-\alpha }{(n-1)^2 h f(\xi _{\beta }) } \sum _{k=1}^n \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )]^2 + o_p(1) \nonumber \\= & {} \frac{- n\alpha }{(n-1)^2 h f(\xi _{\beta }) } \int _{-\infty }^{\infty } \omega \left( \frac{\alpha {{\hat{\xi }}}_{\beta } - x}{h}\right) [I(F(x)\le \beta )-\beta ]^2\,\mathrm{d}F(x) + o_p(1) \nonumber \\= & {} o_p(1),\end{aligned}$$

(30)

$$\begin{aligned} M_2= & {} \frac{1}{n-1} \sum _{k=1}^n K\left( \frac{\alpha {{\hat{\xi }}}_{\beta } - X_k}{h}\right) [ I (X_k \le \xi _{\beta }) - F_n(\xi _\beta )] \nonumber \\= & {} \frac{n}{n-1} \int _{-\infty }^{\infty } K\left( \frac{\alpha \xi _{\beta } - x}{h}\right) [I(F(x)\le \beta )-\beta ]\, \mathrm{d}F(x)+ o_p(1) \nonumber \\= & {} \frac{n}{n-1} \int _{-\infty }^{\infty } \int _{-\infty }^{\frac{\alpha \xi _{\beta } - x}{h}} \omega (y)\,\mathrm{d}y [I(F(x)\le \beta )-\beta ] \mathrm{d}F(x) + o_p(1) \nonumber \\= & {} \int _{-\infty }^{\infty } I(F(x) \le F(\alpha \xi _\beta ))[I(F(x)\le \beta )-\beta ]\, \mathrm{d}F(x) + o_p(1) \nonumber \\= & {} \theta _{\alpha \beta }(1-\beta )+ o_p(1). \end{aligned}$$

(31)

From (29), (30), and (31), we get $J_3=- 2\alpha (1-\beta )\theta _{\alpha \beta } \frac{f(\alpha \xi _{\beta })}{f(\xi _{\beta })} + o_p(1)$. Therefore,

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n {{\hat{V}}}_{k}^2(\alpha , \beta ) \mathop {\longrightarrow }\limits ^{{p}} \frac{ \alpha ^2 \beta (1-\beta ) f^2(\alpha \xi _\beta )}{f^2(\xi _{\beta })} + \theta _{\alpha \beta } - 2\alpha (1-\beta )\theta _{\alpha \beta } \frac{f(\alpha \xi _{\beta })}{f(\xi _{\beta })}.\qquad \end{aligned}$$

(32)

In sum, we have that

$$\begin{aligned} \frac{1}{n} \sum _{k=1}^n \{ {{\hat{V}}}_{k}(\alpha , \beta ) - \theta _{\alpha \beta }\} ^2 \mathop {\longrightarrow }\limits ^{{p}} \sigma ^2_{\alpha \beta }. \end{aligned}$$

The Proof of Theorem 3.1

It follows immediately from Lemmas 1 and 2.

The Proof of Theorem 3.2

Let $g(\lambda )=\frac{1}{n} \sum _{i=1}^n\frac{{\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }}{1+\lambda ({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })}$. It is easy to check that

$$\begin{aligned} 0= & {} |g(\lambda )|=\frac{1}{n}\left| \sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }) - \lambda \sum _{i=1}^n \frac{({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })^2}{1+\lambda ({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })}\right| \nonumber \\\ge & {} \left| \frac{\lambda }{n}\sum _{i=1}^n \frac{({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })^2}{1+\lambda ({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })}\right| - \left| \frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }\right| \nonumber \\\ge & {} \frac{|\lambda |S_n}{1+|\lambda |Z_n} - \left| \frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })\right| , \end{aligned}$$

(33)

where $S_n=\frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })^2$ and $Z_n=\max _{1 \le i \le n}|{\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }|$.

From Lemmas 1 and 2, we have $|\lambda |=O_p(n^{-\frac{1}{2}})$. Put $\gamma _i=\lambda ({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })$, then we have $\max _{1 \le i \le n}|\gamma _i|=o_p(1)$, and

$$\begin{aligned} 0= & {} g(\lambda )=\frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })\left( 1- \gamma _i + \frac{\gamma _i^2}{1+\gamma _i}\right) \nonumber \\= & {} \frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })-S_n\lambda + \frac{\lambda ^2}{n}\sum _{i=1}^n\frac{({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })^3}{1+\gamma _i}\nonumber \\= & {} \frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })-S_n\lambda + o_p(n^{-1/2}), \end{aligned}$$

(34)

which implies that $\lambda =S_n^{-1}\frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }) + \beta _n$, where $\beta _n=o_p(n^{-1/2})$.

Therefore,

$$\begin{aligned} l_n(\theta _{\alpha \beta })= & {} 2\sum _{i=1}^n \log \{ 1+ \lambda ({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })\}\nonumber \\= & {} 2\sum _{i=1}^n\gamma _i -\sum _{i=1}^n\gamma _i^2 + o_p(1) \nonumber \\= & {} 2n\lambda \frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }) - nS_n\lambda ^2 + o_p(1)\nonumber \\= & {} \frac{n\{\frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta }) \}^2}{S_n} - nS_n\beta _n^2 + o_p(1)\nonumber \\= & {} \frac{n\{\frac{1}{n}\sum _{i=1}^n({\hat{V}}_i(\alpha , \beta )-\theta _{\alpha \beta })\}^2}{S_n} + o_p(1) \mathop {\longrightarrow }\limits ^{{d}} \chi ^2(1). \end{aligned}$$

(35)