Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

2000 • Question 5

Question

Math 6010 • Linear Models 2000 • Question 5

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with common density function \(f\). Let \[ \hat {f}_n(t)=\frac {1}{nh}\sum _{1\leq i\leq n}K\left (\frac {t-X_i}{h}\right ), \] where \(K\) is a symmetric density function and \(K(t)=0\) if \(|t|>1\). We assume that \(h=h(n)\to 0\) as \(n\to \infty \). We assume that the second derivative of \(f\) exists and continuous in a neighborhood of \(t\). We use \(\hat {f}\) to estimate \(f\). Compute the bias of \(\hat {f}_n(t)\).

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with common density function \(f\). Let
\[
\hat{f}_n(t)=\frac{1}{nh}\sum_{1\leq i\leq n}K\left(\frac{t-X_i}{h}\right),
\]
where \(K\) is a symmetric density function and \(K(t)=0\) if \(|t|>1\). We assume that \(h=h(n)\to0\) as \(n\to\infty\). We assume that the second derivative of \(f\) exists and continuous in a neighborhood of \(t\). We use \(\hat{f}\) to estimate \(f\). Compute the bias of \(\hat{f}_n(t)\).

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Topic Tags

Covariance and Independence Measure-Theoretic Probability Symmetric Matrices Uniform Integrability and Truncation Methods