A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

Algorithm for the calculation of the electronic structure of molecules by the expanded Hückel method

An algorithm is described for the calculation of quantum-chemical characteristics of molecules by the expanded Hückel method. The method for the calculation of the complete overlap matrix of Slater atomic orbitals of type ns, np, and nd was modified.     

Electrochemical Analogue to the Cinnamic Acid Photodimerization

The dimerization reaction that occurs during electrochemical methoxylation of cinnamic acid is analogous to photodimerization and leads to the formation of -truxillic acid in the following cathodic–anodic process: The radical cation that forms on the anode converts into a biradical on the cathode, which then undergoes dimerization to -truxillic acid with a current efficiency of 63% (in a diaphragmless electrolyzer). The current efficiency rises to 94% when an alternating current is used. In a diaphragm electrolyzer -truxillic acid fails to form.     

Reactions of acyl anions generated from acid chlorides and diiodosamarium

Acid chlorides react with SmI₂ to give α-diketones. Various experiments show that the initially formed acyl radical is rapidly transformed into an acyl anion which is thus generated by quite an unusual route.This species acylates acid chlorides, aldehydes and ketones in situ to give α-diketones and α-ketols. There are some limitations to this acyl anion chemistry but the reactions are realized with a good efficiency at room temperature in THF solution.