2 - Asymptotic expansion and asymptotic distribution of likelihood functions

Summary

Summary

In this chapter, some fundamental results regarding the asymptotic expansion, in the probability sense, and also the asymptotic distribution of certain likelihood functions are derived. These results constitute the backbone of the remaining chapters in this monograph and their derivation rests heavily on the material discussed in Chapter 1.

The underlying probability model involved in our discussions is that of a Markov process satisfying certain reasonable regularity conditions. This model includes, of course, as a special but important case the model consisting of independent identically distribution (i.i.d.) random variables (r.v.s) which is assumed more often in statistical literature.

We now proceed to present a brief outline of what is done in this chapter, since the various derivations are rather involved and the reader might lose sight of the essence of the results. In Section 2, we gather together the various assumptions which are used in the present chapter and which also are basic for what is discussed in the subsequent chapters. The new element here is the assumption of differentiability in quadratic mean of the square root of the probability density function. It replaces the assumption usually made in statistical literature about the existence of two or three pointwise derivatives of the logarithm of the density. As is shown by LeCam [6], the classical Cramér type assumptions imply the one made here. The underlying conditions are then verified in a number of examples which are used throughout this monograph for illustrative purposes.