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AuthorIbragimov, I. A. author
TitleStatistical Estimation [electronic resource] : Asymptotic Theory / by I. A. Ibragimov, R. Z. Has'minskii
ImprintNew York, NY : Springer New York : Imprint: Springer, 1981
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Descript VII, 403 p. online resource


when certain parameters in the problem tend to limiting values (for example, when the sample size increases indefinitely, the intensity of the noise apยญ proaches zero, etc.) To address the problem of asymptotically optimal estimators consider the following important case. Let X 1, X 2, ... , X n be independent observations with the joint probability density !(x,O) (with respect to the Lebesgue measure on the real line) which depends on the unknown patameter o e 9 c R1. It is required to derive the best (asymptotically) estimator 0:( X b ... , X n) of the parameter O. The first question which arises in connection with this problem is how to compare different estimators or, equivalently, how to assess their quality, in terms of the mean square deviation from the parameter or perhaps in some other way. The presently accepted approach to this problem, resulting from A. Wald's contributions, is as follows: introduce a nonnegative function w(0l> ( ), Ob Oe 9 (the loss function) and given two estimators Of and O! n 2 2 the estimator for which the expected loss (risk) Eown(Oj, 0), j = 1 or 2, is smallest is called the better with respect to Wn at point 0 (here EoO is the expectation evaluated under the assumption that the true value of the parameter is 0). Obviously, such a method of comparison is not without its defects


Basic Notation -- The Problem of Statistical Estimation -- Local Asymptotic Normality of Families of Distributions -- Properties of Estimators in the Regular Case -- Some Applications to Nonparametric Estimation -- Independent Identically Distributed Observations. Densities with Jumps -- Independent Identically Distributed Observations. Classification of Singularities -- Several Estimation Problems in a Gaussian White Noise

Mathematics Probabilities Mathematics Probability Theory and Stochastic Processes


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