Spline Estimators of the Density Function of a Variable Measured with Error

Abstract

The estimation of the distribution function of a random variable X measured with error is studied. It is assumed that the measurement error has a normal distribution with known parameters. Let the i-th observation on X be denoted by Y_i=X_i+ε_i , where ε_i is the measurement error. Let {Y_i } ( i=1, 2, …, n) be a sample of independent observations. It is assumed that {X_i } and {ε_i } are mutually independent and each is identically distributed. The proposed estimator is a spline function that transforms X into a standard normal variable. The parameters of the spline function are obtained by maximum likelihood estimation. The number of parameters is determined by the data with a simple criterion, such as AIC. Computationally, a weighted quantile regression estimator is used as the starting value for the nonlinear optimatization procedure of the MLE. In a simulation study, both the quantile regression estimator and the maximum likelihood estimator dominate an optimal kernel estimator and a mixture estimator under a wide class of scenarios.

Keywords:

• Cubic spline
• Kernel estimation
• Maximum likelihood estimation
• Normal transformation
• Mixture of normals
• Quantile regression

Acknowledgments

This research was supported by the Iowa Soybean Promotion Board as part of the project “On-Farm Site-Specific Crop Management for Iowa.” Computing for the research was done with equipment purchased with funds provided by an NSF SCREMS grant award DMS9707740.