A simple solution to a nonparametric maximum likelihood estimation problem
The nonparametric maximum likelihood (ML) approach in estimating a density function f encounters, among others, the difficulty that the likelihood \[ L(f)=\prod^{n}_{i=1}f(x_ i)=\prod^{n}_{i=1}<\delta_{x_ i},f>, \] where \(x_ 1,...,x_ n\) are i.i.d. with density function f, has no finite maximum as f is allowed to range over an appropriate space of densities. One way to overcome this difficulty is to modify the linear functional \(<\delta_{x_ i},f>\) in some suitable way. The authors propose three approaches. The first is based on the method of sieves. The second method is to replace the functional by an inner product in a Hilbert space which contains both \(\delta\) and f. The third method is a standard ML method with the additional condition that \(D^ 2f\) belong to a bounded set. The authors consider one specific sieve (a double exponential convolution sieve) and show that the first method includes the other two, and they all lead to the problem of maximizing the same function. The authors then obtain a closed-form expression for the ML estimator in this case and show that it is consistent.
scientific article; zbMATH DE number 3949497