Adaptive Density Estimation From Data With Small Measurement Errors

In this paper, we study the problem of density estimation from data that contains small measurement errors. The only assumption on these errors is that the maximal measurement error is bounded by some real number converging to zero for sample size tending to infinity. In particular, we do not assume that the measurement errors are independent with expectation zero. We estimate the density by a standard kernel density estimate applied to data with measurement errors and derive a data-driven method to choose its bandwidth. We derive an adaptation result for this estimate and analyze the expected L₁ error of our density estimate depending on the smoothness of the density and the size of the maximal measurement error. The results are applied in a density estimation problem in a simulation model, where we show under suitable assumptions that the L₁ error of our newly proposed estimate converges to zero much faster than the L₁ error of the standard kernel density estimate if both are based on the same number of observations in the simulation model. The performance of the method in case of finite sample size is analyzed using simulated data.