Unbiased Estimators and Classification Problems for Multivariate Normal Populations

Abusev, R. A.; Lumel'skii, Ya. P.

doi:10.1137/1125048

Cited by 14 publications

(8 citation statements)

References 3 publications

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“…A result similar to Theorem 1 was presented in other terms in [30,31]. A result similar to Theorem 1 was presented in other terms in [30,31].…”

supporting

confidence: 63%

Markov random walks in the space of a sufficient statistic and statistical inference

Lumel'skii¹

1998

J Math Sci

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skii (Perm, Russia) UDC 519.2 Sequential schemes of observations play an important role in the modern theory of statistical inference. In these schemes, the time at which observations are terminated is a random variable. Sequential schemes were proposed by A. Wald [1] and, based on the likelihood ratio, were oriented to testing hypotheses. This direction of sequential analysis is well known. Its further development was considered in [2,3]. Another important direction of sequential analysis based on the plans of random walks turned out to attract less attention. The paper of Girshick, Mosteller, and Savage [4] was the first in this direction. This direction of sequential analysis has been partially described in [5, Chap. 12; 6, w and w 7, w w Chap. 4]. The aim of this review is to describe the models of sequential analysis based on the plans of random walks. Here we consider some results concerning the models of random walks. Much attention will be paid to the research concerning the construction of random walk models performed at Perm State University in 1970-1995 by Ya. P. Lumel'skii, Ya. M. L'vovskii, S. I. Frolov, E. G. Tsylova, V. V. Chichagov, and others.At present, the following basic schemes of random walks and some of their generalizations are known. 1. Binomial and polynomial schemes of random walks (see [4-5, 8, 9-10]). 2. Hypergeometric random walks for the first time considered in [11] (two-dimensional case), [12] (multivariate case). The random walks that arise in problems connected with the estimation of the size of a finite population were proposed in [6, ~5.41. 3. P61ya random walks, including polynomial and multivariate hypergeometric ones, were first constructed in [13] and were also considered in [7, 14]: Generalizations of the Pdlya random walks can be found in [15-18]. 4. Poisson random walks: one-dimensional [19], multivariate [20]. Generalizations of Poisson random walks were considered in [21, 22]. 5. Random walks connected with homogeneous [23-25] and nonhomogeneous [26] discrete-time Markov chains were extended to continuous time in [27]. 6. Wiener random walks: one-dimensional [5, 19] and multivariate [28, 31], 7. Negative-binomial random walks [5, 19, 29].The plans of random walks mentioned above make it possible to extend in some sense the families of distributions known earlier. For these families the theory of statistical estimation has been developed. Much attention has been paid to the construction of unbiased estimators expressed through a sufficient statistic, that is, through the coordinates of a boundary point.In all the schemes mentioned above, the random walk takes place in the space of a sufficient statistic. Consider the following rather general model of Markov random walks. Basic definitions and concepts correspond to those in [5,7] applied to first-passage plans.Consider a Markov random walk starting from the origin O over points of some space W (O E W). In the space W a stopping boundary G C W and a boundary point F E G of the random walk are assumed to be given. The ...

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“…A result similar to Theorem 1 was presented in other terms in [30,31]. A result similar to Theorem 1 was presented in other terms in [30,31].…”

supporting

confidence: 63%

Markov random walks in the space of a sufficient statistic and statistical inference

Lumel'skii¹

1998

J Math Sci

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“…Abusev and Lumelsky [6] had taken into consideration in the problem of classification not a single observation but a group χ ο = (soi>· · · > X ON O ) °^ ^° independent observations. Such problems arise in a medical and technical diagnostics, in particular, in epidemiology and quality control.…”

Section: Discriminant Analysismentioning

confidence: 99%

“…For N ^ NQ -f p + k and Ni > NQ, the unique minimum variance unbiased estimator of the density function fi(xo,Ao) given by(6) with parameters μι and Σ is of the form r./. , NiN<> ,... Λ1 (Ν-ΛΓ β -* [ψ (A -Ao -N __ NQ (*o -xi)(xo -x<) jj where N iNo 'The proof of the above theorem is the same as that of Theorem 2.…”

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confidence: 99%

Statistical group classification rules for the multivariate Student t-distribution

Krzyśko¹,

Wołyński²

1993

Random Operators and Stochastic Equations

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“…If the objects in the classes are independent and normally distributed, then one can apply the results from [1][2][3][4][5].…”

mentioning

confidence: 99%

Bayes estimation and group classification for multivariate dependent observations

Abusev¹,

Mazanova²

1996

J Math Sci

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Unbiased Estimators and Classification Problems for Multivariate Normal Populations

Cited by 14 publications

References 3 publications

Markov random walks in the space of a sufficient statistic and statistical inference

Markov random walks in the space of a sufficient statistic and statistical inference

Statistical group classification rules for the multivariate Student t-distribution

Bayes estimation and group classification for multivariate dependent observations

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