Math 6010 • Linear Models

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with distribution function \(F\) and density function \(f\). We assume that the second derivative of \(f\) is bounded. We estimate \(f\) by
\[
f_n(t)=\frac{F_n(t+h_n)-F_n(t-h_n)}{2h_n},
\]
where \(F_n\) is the empirical distribution function and \(h_n\to0\) as \(n\to\infty\).
\begin{parts}
\item Show that \(Ef_n(t)\to f(t)\) as \(n\to\infty\). (5 points)
\item Show that if \(nh_n\to\infty\), then
\[
\frac{f_n(t)-Ef_n(t)}{(\operatorname{var}(f_n(t)))^{1/2}}
\]
is asymptotically standard normal as \(n\to\infty\). (5 points)
\end{parts}