Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

1993 • Question 9

Question

Math 6010 • Linear Models 1993 • Question 9

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with density function \(f\). We assume that \(f'\) is bounded. Let \(K\) be a function satisfying \(\int _{-\infty }^{\infty }K(u)\,du=1\), \(K'(u)\) is bounded and \(K(u)=0\), if \(|u|\geq a\) where \(a\) is a constant. The density \(f\) is estimated by \[ \hat f_n(t)=\frac {1}{nh}\sum _{1\leq i\leq n}K\left (\frac {t-X_i}{h}\right ). \]

Show that \(\hat f_n(t)\) is an almost surely uniformly consistent estimator for \(f\) on \([\alpha ,\beta ]\), \(-\infty <\alpha <\beta <\infty \).

Let \(X_1,X_2,\ldots,X_n\) be independent identically distributed random variables with density function \(f\). We assume that \(f'\) is bounded. Let \(K\) be a function satisfying \(\int_{-\infty}^{\infty}K(u)\,du=1\), \(K'(u)\) is bounded and \(K(u)=0\), if \(|u|\geq a\) where \(a\) is a constant. The density \(f\) is estimated by
\[
\hat f_n(t)=\frac{1}{nh}\sum_{1\leq i\leq n}K\left(\frac{t-X_i}{h}\right).
\]

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Show that \(\hat f_n(t)\) is an almost surely uniformly consistent estimator for \(f\) on \([\alpha,\beta]\), \(-\infty<\alpha<\beta<\infty\).

Topic Tags

Convergence in Probability and Almost Sure Convergence Covariance and Independence Estimator Distributions Uniform Integrability and Truncation Methods