A penalty method for nonparametric estimation of the logarithmic derivative of a density function
This paper deals with the estimation of the function \(\psi_ 0=-f'\!_ 0/f_ 0\) from independent random variables \(X_ 1,...,X_ n\) with common unknown density \(f_ 0\) and cumulative distribution function \(F_ 0\). From the remark that, under mild conditions, \(\psi_ 0\) is the unique minimizer of \(\int [\psi^ 2-2\psi ']dF_ 0\) the author proposes as an estimate of \(\psi_ 0\) the function \(\psi_{n,\lambda}\) minimizing \[ \lambda \int [L\psi]^ 2+\int [\psi^ 2-2\psi ']dF_ n \] where \(F_ n\) is the empirical distribution function and L is a linear differential operator. Suitable assumptions involve (a.s.) the existence and uniqueness of \(\psi_{n,\lambda}\); a linear system is given which yields this estimate. The case \(L=d^ 2/dx^ 2\) gives the maximum likelihood normal estimate as the limit of \(\psi_{n,\lambda}\) for \(\lambda\) tending to infinity. Asymptotics under periodicity assumptions indicate that the estimator is consistent and achieves the optimal rate of convergence.
scientific article; zbMATH DE number 3925997