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Prerequisites for the publication contain senior undergraduate/beginning graduate-level classes in likelihood and statistics.

Readership: Graduate scholars and study mathematicians drawn to mathematical records

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**Sample text**

X(Oo) + on(1) [((On - Oo) 2 +(On - 01) 2 ) K(Tn :=::; Tn) J ~ 1) J, ~ 0, > n 2 (01 - Oo) 2 lP'O(T. X( Oo)} as n --+ oo. 8. The rate of convergence may be different from O(n- 1 ) for some other irregular statistical experiments, but those models are not asymptotically exponential. 8 is not regular (the Fisher information does not exist) if -1 < a :::; 1. The rate of convergence in this model depends on a and is, generally speaking, different from 0 (n - 1 ). 5. Sharp Lower Bound ! 7 is far from being sharp.

In this section, we prove the sharp lower bound with r * = r* = 1 for the normal observations. To do this, we have to overcome the same technical difficulties and we will need the same ideas as in the case of more general observations discussed in the next section. 8. 4, for any estimator Bn, the following lower bound holds: liminf n2 sup lEe n--+oo (J eEe [(en- B) 2 ] ;::: 1. Proof. 4, we can take (}" 2 = 1. fii] where b will be chosen later. Under the assumption on 8, it contains this interval for all sufficiently large n.

Find the weighted posterior mean of B. 10. Consider a set of independent observations X~, ... , Xn "' N(B, a- 2 ), where B is assumed random with the prior density N(J-t, a-~). Show that the weighted posterior distribution of B is also normal with the mean (na-~ Xn + J-ta- 2 )/(na-~ + a- 2) and variance a- 2 a-V(na-~ + a- 2). Note that the family of normal distributions is self-conjugate. 11. Find a conjugate distribution and the corresponding Bayes estimator for the parameter B in the exponential model with p(x, B) = Bexp{-Bx}, x,B > 0.