Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2007 • Question 3

Question

Math 6010 • Linear Models Fall 2007 • Question 3

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with distribution function \(F\). We use \[ \hat f_n(t)=\frac {1}{nh}\sum _{1\leq i\leq n} K\left (\frac {t-X_i}{h}\right ) \] to estimate \(f=F'\).

We assume that the following conditions are satisfied: \[ \int _{-1}^{1}K(u)\,du=1, \qquad K(t)=0,\quad \text {if } t\notin [-1,1]. \] \(F\) is three times differentiable and the derivatives are uniformly bounded on the real line \[ h=h(n)\to 0. \]

(i) Compute the bias of \(\hat f_n(t)\) for any \(t\) and any \(n\). (7 points)

(ii) Prove that \(\hat f_n(t)\) is asymptotically unbiased for any \(t\) as \(n\to \infty \). (7 points)

(iii) Compute the mean square error of \(\hat f_n(t)\) (7 points)

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed
random variables with distribution function \(F\). We use
\[
  \hat f_n(t)=\frac{1}{nh}\sum_{1\leq i\leq n}
  K\left(\frac{t-X_i}{h}\right)
\]
to estimate \(f=F'\).

We assume that the following conditions are satisfied:
\[
  \int_{-1}^{1}K(u)\,du=1,
  \qquad
  K(t)=0,\quad\text{if } t\notin[-1,1].
\]
\(F\) is three times differentiable and the derivatives are uniformly bounded
on the real line
\[
  h=h(n)\to0.
\]

(i) Compute the bias of \(\hat f_n(t)\) for any \(t\) and any \(n\). (7 points)

(ii) Prove that \(\hat f_n(t)\) is asymptotically unbiased for any \(t\) as
\(n\to\infty\). (7 points)

(iii) Compute the mean square error of \(\hat f_n(t)\) (7 points)

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Topic Tags

Covariance and Independence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods