In the non-parametric density estimation, one is interested in estimating the unknown density f based on an i.i.d. sample X_1, ..., X_n, drawn from f. In practice, one has rarely access to the direct data X_1, ..., X_n due to measurement errors. In fact, a very widely discussed model is the model of additive measurement errors where one only has access to the data Y_i= X_i +e_i, where e_i, denotes the measurement error. In the past decades the model of multiplicative measurement errors, that is, one has only access to Y_i =X_i U_i, was solved by application of the logarithm to reduce the model to an additive measurement error model. Recent developments have shown that one can instead use the Mellin transform to solve the upcoming inverse problem directly.
In this talk, I start by introducing the general statistical framework. Then I will define the Mellin form in a functionalanalytical context and discuss the corresponding Sobolev spaces. In the last third of my talk, I will present our research results in the density estimation under multiplicative measurement errors.