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**On the asymptotic theory of estimation when the limit of the log- likelihood ratios is mixed normal.**
*(English)*
Zbl 0584.62042

Some results concerning the asymptotic theory of estimation are presented when the limit distribution of the log-likelihood ratios is mixed normal. More specifically, the notion of a locally asymptotically mixed normal (LAMN) sequence of families of distributions is introduced, and it is shown that when a certain kind of differentiability in quadratic mean type regularity condition is satisfied the given sequence of families satisfies the LAMN condition.

As a consequence of the LAMN condition, it is shown that the limit distribution of a regular sequence of estimators can be conditionally decomposed as a convolution. Using this conditional convolution result, some results concerning the asymptotic lower bound for risk functions of estimators are obtained.

Given that the sequence of families satisfies the LAMN condition quite general additional assumptions are sought under which it is shown that the maximum probability estimators, maximum likelihood estimators and a certain class of Bayes estimators are asymptotically optimal in a certain sense. A result concerning the posterior approximation at the true value of the parameter is also presented.

As a consequence of the LAMN condition, it is shown that the limit distribution of a regular sequence of estimators can be conditionally decomposed as a convolution. Using this conditional convolution result, some results concerning the asymptotic lower bound for risk functions of estimators are obtained.

Given that the sequence of families satisfies the LAMN condition quite general additional assumptions are sought under which it is shown that the maximum probability estimators, maximum likelihood estimators and a certain class of Bayes estimators are asymptotically optimal in a certain sense. A result concerning the posterior approximation at the true value of the parameter is also presented.

### MSC:

62F12 | Asymptotic properties of parametric estimators |

62E20 | Asymptotic distribution theory in statistics |