Math 6010 • Linear Models

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with distribution function \(F\). We assume that \(f=F'\) has a bounded second derivative. \(K(t)\) is a non-negative function satisfying
\[
K(t)=K(-t)\quad\text{(symmetric),}
\]
\[
\int_{-\infty}^{\infty}K(t)\,dt=1
\quad\text{and}\quad
K(t)=0\text{ if }t\notin[-1,1].
\]
The sequence \(h=h(n)\) is positive and goes to 0. We estimate \(f\) with
\[
\hat{f}_n=\frac{1}{nh}\sum_{1\leq i\leq n}K\left(\frac{t-X_i}{h}\right).
\]
\begin{enumerate}
\item Compute \(B_n\), the bias of \(\hat{f}_n\). (10 points)
\item Compute the limit of \(B_n/h^2(n)\) as \(n\to\infty\). (10 points)
\end{enumerate}