A semiparametric estimator of the distribution function of a variable measured with error

The estimation of the distribution function of a random variable X measured with error is studied. Let the i-th observation on X be denoted by Y-i = X-i + epsilon(i), where epsilon(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 {epsilon(i)} are mutually independent and each is identically distributed. As is standard in the literature for this problem, the distribution of epsilon is assumed known in the development of the methodology. In practice, the measurement error distribution is estimated from replicate observations. The proposed semiparametric estimator is derived by estimating the quantiles of X on a set of n transformed Y-values and smoothing the estimated quantiles using a spline function. The number of parameters of the spline function is determined by the data with a simple criterion, such as AIC. In a simulation study, the semiparametric estimator dominates an optimal kernel estimator and a normal mixture estimator for a wide class of densities. The proposed estimator is applied to estimate the distribution function of the mean pH value in a field plot. The density function of the measurement error is estimated from repeated measurements of the pH values in a plot, and is treated as known for the estimation of the distribution function of the mean pH value.