Kozhushko, Eh. I.; Sakhanenko, A. I. Conditions of asymptotic efficiency of recursion estimates of the shift parameter. (English. Russian original) Zbl 0646.62025 Sib. Math. J. 28, No. 3, 458-463 (1987); translation from Sib. Mat. Zh. 28, No. 3(163), 133-139 (1987). Let \(Y_ 1,Y_ 2,...,Y_ n\) be i.i.d. random variables with a density \(p(y-\theta)\), where \(\theta \in R\) is an unknown parameter and P has a finite Fisher information \(I(p).\) The authors study recursive estimates \(\theta^*_ n\), which are asymptotically efficient. The sequence \(\theta^*_{n+1}=\phi_{n+1}(\theta^*_ n,Y_{n+1})\) of estimators suggested by the authors satisfies the following relations \(\forall y:\) \[ P_{\theta}\{(nI(p))^{1/2}(\theta^*_ n- \theta)<y\}\to \Phi (y), \]\[ M_{\theta}[f((nI(p))^{1/2}(\theta^*_ n-\theta))]\to \int_{R^ 1}f(y)d\Phi(y), \] where f is a continuous function and for some \(K>0\): \(\sup_{y}| f(y)| /(1+| y|^ K)<\infty\). This theorem generalizes a result of M. B. Nevel’son [Theory Probab. Appl. 25, 569-579 (1981); translation from Teor. Veroyatn. Primen. 25, 577-587 (1980; Zbl 0444.62041)]. Reviewer: V.Olman MSC: 62F12 Asymptotic properties of parametric estimators 62E20 Asymptotic distribution theory in statistics 62L12 Sequential estimation Keywords:maximum likelihood estimate; finite Fisher information; recursive estimates Citations:Zbl 0495.62035; Zbl 0444.62041 PDFBibTeX XMLCite \textit{Eh. I. Kozhushko} and \textit{A. I. Sakhanenko}, Sib. Math. J. 28, No. 3, 458--463 (1987; Zbl 0646.62025); translation from Sib. Mat. Zh. 28, No. 3(163), 133--139 (1987) Full Text: DOI References: [1] A. A. Borovkov, Mathematical Statistics [in Russian], Nauka, Moscow (1984). · Zbl 0575.62002 [2] M. B. Nevel’son, ?Asymptotically efficient recursion estimation of the shift parameter,? in: Probability Theory and Its Applications [in Russian], Vol. 25, No. 3 (1980), pp. 577-587. [3] A. A. Borovkov, Probability Theory [in Russian], Nauka, Moscow (1972). [4] M. B. Nevel’son and R. Z. Khas’minskii, Stochastic Approximation and Recursion Estimation [in Russian], Nauka, Moscow (1972). [5] V. Fabian, ?On asymptotic normality in stochastic approximation,? Ann. Math. Stat.,39, 1327 (1968). · Zbl 0176.48402 · doi:10.1214/aoms/1177698258 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.