Spline estimators 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. It is assumed that the measurement error is normally distributed with mean zero and known variance σ 2. The random variable X is assumed to have a continuous density function. Let the i-th observation on X be denoted by $Yi=Xi+ei$, where $ei$, is the measurement error. Let Y_i ( i = 1,2,…,n) be a sample of independent observations. It is assumed that X_i and $ei$ are mutually independent and each is identically distributed.; Three spline estimators of the density function of X are proposed. The first is a weighted quantile regression spline obtained by acting as if the observations were generated by a set of n X*-values. The n X*-values are defined as a transformation of the original Y-values such that the mean and variance of the X* are consistent estimators of the mean and variance of X. The other two procedures estimate the parameters of the spline function by maximum likelihood. Both use the quantile regression spline estimator as an initial estimator. In all cases, the number of parameters of the spline function is determined by the data with a sample criterion, such as AIC. The proposed spline estimators perform better than a normal mixture estimator and better than a kernel estimator in a simulation study.