Statistical Estimation: Asymptotic Theory by I.A. Ibragimov, R.Z. Has'minskii, S. Kotz

By I.A. Ibragimov, R.Z. Has'minskii, S. Kotz

While sure parameters within the challenge are inclined to restricting values (for instance, while the pattern measurement raises indefinitely, the depth of the noise ap­ proaches 0, etc.) to deal with the matter of asymptotically optimum estimators reflect on the next very important case. enable X 1, X 2, ... , X n be self reliant observations with the joint likelihood density !(x,O) (with recognize to the Lebesgue degree at the genuine line) which depends upon the unknown patameter o e nine c R1. it truly is required to derive the easiest (asymptotically) estimator 0:( X b ... , X n) of the parameter O. the 1st query which arises in reference to this challenge is find out how to evaluate varied estimators or, equivalently, how you can investigate their caliber, when it comes to the suggest sq. deviation from the parameter or maybe in another approach. The shortly approved method of this challenge, because of A. Wald's contributions, is as follows: introduce a nonnegative functionality w(0l> ( ), Ob Oe nine (the loss functionality) and given estimators Of and O! n 2 2 the estimator for which the anticipated loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is named the higher with admire to Wn at aspect zero (here EoO is the expectancy evaluated less than the belief that the real worth of the parameter is 0). evidently, this sort of approach to comparability isn't really with out its defects.

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Example text

In the following examples a sequence of independent identically distributed observations X 10 X 2, ••• , XII' taking on values in Rk and possessing in Rk probability density f(x; 0) with respect to some measure v is considered. (x - 0)2}, 0 E E> = Rl. Denote by 1k an estimator which is Bayesian with respect to the normal distribution Ak with mean 0 and variance (J~ = k. Since the loss function is quadratic, then (see Section 2) f {I {I exp - - I 1 1~2} . 1 that X is a minimax estimator. Consequently, also the equivariant estimator X is optimal in the class of equivalent estimators.

In p(X; 0) = 0, i = 1, ... 6) I To prove the consistency of the maximum likelihood estimators it is convenient to utilize the following simple general result. 1. Let 8. (X·; e) correspond to these experiments. Set Z. (x·; e), u E U = e - e. Then in order t~t the maximum likelihood estimator 0. ){sup Z •. eCu) ~ 1} lui> Y If the last relation is uniform in 0 E K ~ consistent in K. = O. - PROOF. Set U. P~){IO. 1 :::;; -+ ,"'0 O. e, then the estimator O. is uniformly = supu ZeCu). (u) ~ 1}. 5) possesses a solution.

Denote by 1k an estimator which is Bayesian with respect to the normal distribution Ak with mean 0 and variance (J~ = k. Since the loss function is quadratic, then (see Section 2) f {I {I exp - - I 1 1~2} . 1 that X is a minimax estimator. Consequently, also the equivariant estimator X is optimal in the class of equivalent estimators. We note immediately that X is admissible as well (it is more convenient to present a proof of this fact in Section 7). Hence in the problem under consideration the estimator X of parameter (J has all the possible virtues: it is unbiased, admissible, minimax, and equivariant.

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