Kernel Density Estimation with Berkson Error

• Activity Number: 98
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): James Long*+
• Companies: Texas A&M

Abstract

Given a sample $\{X_i\}_{i=1}^n$ from $f_X$, we construct kernel density estimators for $f_Y$, the convolution of $f_X$ with a known error density $f_{\epsilon}$. This problem is known as density estimation with Berkson error and has applications in epidemiology and astronomy. We derive an asymptotic approximation to the mean integrated squared error (MISE) of a kernel density estimator of $f_Y$ as a function of the bandwidth parameter. Using these asymptotic rates we determine the optimal rate of convergence for the bandwidth. We compare MISE rates for several bandwidth selection strategies both asymptotically and at finite samples. The finite sample results demonstrate the importance of smoothing when the error term $f_{\epsilon}$ is concentrated near 0. We apply our methodology to $NO_2$ exposure data.