Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1998 • Question 6

Question

Math 6010 • Linear Models 1998 • Question 6

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 \(h_n\to 0\) as \(n\to \infty \).

(a) Show that \(\E f_n(t)\to f(t)\) as \(n\to \infty \). (5 points)

(b) Show that if \(nh_n\to \infty \), then \[ \frac {f_n(t)-\E f_n(t)}{(\operatorname {Var}(f_n(t)))^{1/2}} \] is asymptotically standard normal as \(n\to \infty \). (5 points)

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 \(h_n\to 0\) as
\(n\to\infty\).

(a) Show that \(\E f_n(t)\to f(t)\) as \(n\to\infty\). (5 points)

(b) Show that if \(nh_n\to\infty\), then
\[
\frac{f_n(t)-\E f_n(t)}{(\var(f_n(t)))^{1/2}}
\]
is asymptotically standard normal as \(n\to\infty\). (5 points)

Topic Tags

Central Limit Theorems Covariance and Independence Multivariate Normal Distributions Probability Inequalities and Tail Bounds