The standard regularity conditions for maximum likelihood estimation involves the existence of two or three derivatives of the density function, as well as domination properties that allow passing the derivative under the integral sign. Le Cam noted that these conditions are too stringent. For example, the derivatives may fail to exist at a point that depends on the parameter, and they require us to work with the "correct" version of the density function. On a more practical level, it might be hard to find functions dominating the first and second derivatives of the log-density. Le Cam proposed replacing the standard conditions by a simpler differentiability in quadratic mean (DQM) assumption. The DQM assumption only involves a first derivative of the density in a weaker sense, and also steer clear of the domination assumptions on the first and second derivatives. In this talk we look at three applications of the DQM theory. The DQM was originally developed for situations where one is "on the model", that is, situations where the assumed model matches the data generating process. In a first part of the talk, we therefore look at a DQM-type theory outside model conditions, i.e., when the assumed parametric model does not match the data generating process. In the second part, the DQM theory is extended to conditional densities and dependent regression data. In the third application we look at how the DQM condition plays a crucial role in the context of parametric approximations to semiparametric models.
The talk builds on joint work with Dennis Christensen, Nils Lid Hjort, Adam Lee, Jonas Moss, and Per Mykland.