Information in semiparametric mixtures of exponential families

In a class of semiparametric mixture models, the score function (and conseq uently the effective information) for a finite-dimensional parameter can be made arbitrarily small depending upon the direction taken in the parameter space. This result holds for a broad range of semiparametric mixtures over exponential families and includes examples such as the gamma semiparametri c mixture, the normal mean mixture, the Weibull semiparametric mixture and the negative binomial mixture. The near-zero information rules out the usua l parametric root n rate for the finite-dimensional parameter, but even mor e surprising is that the rate continues to be unattainable even when the mi xing distribution is constrained to be countably discrete. Two key conditio ns which lead to a loss of information are the smoothness of the underlying density and whether a sufficient statistic is invertible.