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Teoriya Veroyatnostei i ee Primeneniya, 1974, Volume 19, Issue 4, Pages 700–713
(Mi tvp3974)
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This article is cited in 7 scientific papers (total in 7 papers)
On sequentional estimation of the location parameter for families of distributions with discontinuous densities
I. A. Ibragimova, R. Z. Khas'minskiib a Moscow
b Leningrad
Abstract:
We consider sequential estimation of the location parameter $\theta$ from independent observations $X_1,X_2,\dots$ with a common probability density function $f(x-\theta)$; $x,\theta\in R^1$.
Under the conditions:
(i) the only discontinuities of $f(x)$ are jumps at points $x_1,\dots,x_r$,
(ii) $\displaystyle{\int_{-\infty}^\infty|f'(x)|\,dx<\infty}$,
(iii) $\displaystyle{\biggl(\sum_if^2(x_i+0)\biggr)\biggl(\sum_if^2(x_i-0)\biggr)>0}$,
we construct two invariant sequential procedures $[d,\tau]$, $\mathbf E_\theta\tau\le n$, such that
$$
\varlimsup_n\mathbf E_\theta|d_\tau-\theta|^a/\mathbf E_\theta|\widetilde t_n-\theta|^a<1,\quad a>1,
$$
and $\widetilde t_n$ is the best invariant estimator of $\theta$ corresponding to the loss function $|u-\theta|^a$.
Received: 04.05.1973
Citation:
I. A. Ibragimov, R. Z. Khas'minskii, “On sequentional estimation of the location parameter for families of distributions with discontinuous densities”, Teor. Veroyatnost. i Primenen., 19:4 (1974), 700–713; Theory Probab. Appl., 19:4 (1975), 669–682
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https://www.mathnet.ru/eng/tvp3974 https://www.mathnet.ru/eng/tvp/v19/i4/p700
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Abstract page: | 211 | Full-text PDF : | 76 |
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