I. A. Ibragimov

Local Asymptotic Normality of Families of Distributions

In a number of interesting papers of Hajek, LeCam, and other authors, it was proved that many important properties of statistical estimators follow from the asymptotic normality of the logarithm of the likelihood ratio for neighborhood hypotheses (for values of parameters close to each other) regardless of the relation between the observations which produced the given likelihood function. This chapter is devoted to an investigation of the conditions under which this property is valid for various models and to corollaries of this property.

Independent Identically Distributed Observations. Densities with Jumps

The estimation theory given in Chapters II and III utilizes to a large extent the regularity of the experiments under consideration. Obviously this is the most important case, however, it is not difficult to give examples of very interesting problems where the regularity condition is not fulfilled.

The Problem of Statistical Estimation

Observations constitute the basis of a statistical experiment: these may be numerical data or data of some other nature obtained as a result of a statistical experiment. The problem is to make, based on these data, some definite and sufficiently reliable conclusions concerning the object under investigation.

Some Applications to Nonparametric Estimation

Nonparametric estimation is a large branch of mathematical statistics dealing with problems of estimating functional or elements of some functional spaces in situations when these are not determined by specifying a finite number of parameters. In this chapter we shall show by means of several examples how the ideas of parametric estimation presented in Chapters I–III can be applied to problems of this kind.

Properties of Estimators in the Regular Case

In the preceding chapter, some general properties of estimators in the case when the family of distributions obeys the LAN property were established. In particular, a minimax lower bound on the quality of various estimates for a large class of loss functions were derived. The main purpose of the present chapter is to prove the asymptotic efficiency of a maximum likelihood estimator and of a large class of Bayesian and generalized Bayesian estimators for regular families of experiments. Evidently, certain new restrictions on the families under consideration will be required.

Independent Identically Distributed Observations. Classification of Singularities

As in the preceding chapter, we shall consider below a sequence of independent, identically distributed observations X j with values in R 1. However, in order to achieve maximal clarity in the exposition, we shall study in this chapter the simplest possible model, that of estimating a location parameter. Extension of the results of this Chapter to the case of a more general model described in Section V.I does not present difficulties and does not lead to substantially new phenomena.

Several Estimation Problems in a Gaussian White Noise

A statistical experiment generated by observing a signal with Gaussian white noise is a very convenient model for investigation. The simple form of the likelihood ratio Z (see formula (2.A.17)) and the normality of the random field ln Z — these facts allow us to avoid many difficulties. At the same time, the model is sufficiently interesting since the signal observed may to a large extent depend arbitrarily on the parameter. Finally, this model is of importance in many applications (see [72]). In this chapter we shall first—in Section 1—study the effects which arise in the case of an unbounded or expanding parameter set Θ, since this case is of special importance in information theory. Next, in Sections 2 and 3 the case when the signal 5 depends on the parameter in a discontinuous manner will be considered. Finally, in Sections 4 and 5 examples of non-parametric problems are given.