Consistent pseudo-maximum likelihood estimators

The development of the literature on the pseudo maximum likelihood (PML) estimators would not have been so efficient without the modern proof of consistency of extremum estimators introduced at the end of the sixties by E. Malinvaud and R. Jennrich. We discuss this proof and replace it in an historical perspective. In this paper we also provide a survey of the literature on consistent (PML) estimators. We emphasize the role of the white noise assumptions on the set of pseudo distributions leading to consistent estimators. The stronger these assumptions, the larger the set of consistent PML estimators. We also illustrate the importance of these PML approaches in big data environment