Density Estimation with Normal Measurement Error with Unknown Variance

Abstract: This paper deals with the problem of estimating a density based on observations which are contaminated by a normally distributed error whose variance is unknown. In the case of a completely unknown error variance, the impossibility of a uniformly consistent estimation is shown; however, a semi-uniformly consistent estimator is constructed under nonparametric smoothness conditions on the target density, and its rates are studied. If, in contrast, the error variance can be located in a known compact interval, we derive uniform consistency for this estimator which achieves nearly optimal rates. Simulations show the practical merit of the estimator.