Suppose $X_1,\cdots,X_n\overset{iid}\sim F$, the $q$th quantile $\theta=F^{-1}(q)$. Let $K$ denote an $r$th order kernel. That is for some integer $r\ge2$ and constant $C\neq0$, $K$ satisfies $$$$ \begin{eqnarray*} \int u^jK(u)du=\left\{ \begin{aligned} 1, & \quad\textrm{if} \quad j=0, \\ 0, & \quad\textrm{if} \quad 1\le j\le r-1,\\ C, & \quad\textrm{if} \quad j=r. \end{aligned} \right. \end{eqnarray*} Let $$w_i=w_i(\theta)=\int_{-\infty}^\theta\frac1hK(\frac{y-X_i}h)dy-q$$ and $$\bar w_2=\frac1n\sum_{i=1}^nw_i^2$$ Show that $$E(\bar w_2)=q(1-q)+o(1).$$
$$E(\bar w_2)=Ew_i^2=E(\int_{-\infty}^\theta\frac1hK(\frac{y-X_i}h)dy)^2-2q\cdot E(\int_{-\infty}^\theta\frac1hK(\frac{y-X_i}h)dy)+q^2$$
By Fubini's theorem and Taylor expansion, I can get the second term equals to $$-2q(q+o(1))=-2q^2+o(1)$$
My question is how to deal with the first term?
This question comes from the paper
Smoothed Empirical Likelihood Confidence Intervals for Quantiles
by Song Xi Chen and Peter Hall on page 1175.