Math 6010 • Linear Models

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 \(|u|\ge a\) where \(a\) is a constant. The density \(f\) is estimated by \[ \hat f_n(t)=\frac {1}{nh}\sum _{1\le i\le n} K\left (\frac {t-X_i}{h}\right ). \] Show that \(\hat f_n(t)\) is an almost surely uniformly consistent estimator for \(f\) on \([\alpha ,\beta ]\), \(-\infty <\alpha <\beta <\infty \).