Density Deconvolution with Replicate Measurements and Auxiliary Data

Abstract

We present two deconvolution estimators for the density function of a random variable X that is measured with error. The first estimates the density of X from the set of independent replicate measurements W[subscript r,j], where W[subscript r,j]=X[subscript x]+U[subscript r,j] for r=1,...,n and j=1,...m[subscript r]. We derive an estimator assuming that the U[subscript r,j] are normally distributed measurement errors having unknown and possibly nonconstant variances σ[subscript r]². The estimator generalizes the deconvolution estimator of Stefanski and Carroll (1990), with the measurement error variances estimated from replicate observations. We derive the integrated meansquared error and examine the rate of convergence as n → ∞ and the number of replicates is fixed.The finite-sample performance of the estimator is illustrated through a simulation study and an example.

The second is a semi-parametric deconvolution estimator that assumes the availability of a covariate vector Z statistically related to X, but independent of the error in measuring X, and such that the regression error X-E(X|Z) is normally distributed. The estimator combines parametric modeling of the regression residuals with nonparametric estimation of the mean function. The asymptotic properties of the estimator are discussed. The reliance of the estimator on assumptions of the regression model and normality of model errors is examined via simulation, and an application to real data is presented.