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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
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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
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