Some aspects of minimum variance unbiased estimation in presence of ancillary statistics

Nayak, Tapan K.; Sinha, Bimal K.

doi:10.1016/j.spl.2012.02.024

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…Applied to the Nile problem, Theorem 1 proves that a statistic satisfying rather general "regularity type" conditions (Conditions 1, 2, 3a, b) is not a UMVUE (the existence of a nonconstant UMVUE is an open problem, according to Nayak and Sinha [35]).…”

Section: Statistical Methods and The Original Nile Problemmentioning

confidence: 99%

“…See also Hinkley [17] and Kariya [28] for related results. According to Nayak and Sinha [35], the problem of existence of UMVUEs is open.…”

Section: Problems Closely Related To the Nile Problemmentioning

confidence: 99%

“…Due to incompleteness of the minimal sufficient statistic, the existence and construction of UMVUEs do not follow from the Rao-Blackwell and Lehmann-Scheffé theorems and become a nontrivial problem which requires a new approach. Nayak and Sinha [35] mentioned the existence of UMVUEs in the models (1)…”

Section: Introductionmentioning

confidence: 99%

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On the Nile problem by Sir Ronald Fisher

Kagan¹,

Malinovsky²

2013

Electron. J. Statist.

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The Nile problem by Ronald Fisher may be interpreted as the problem of making statistical inference for a special curved exponential family when the minimal sufficient statistic is incomplete. The problem itself and its versions for general curved exponential families pose a mathematical-statistical challenge: studying the subalgebras of ancillary statistics within the σ-algebra of the (incomplete) minimal sufficient statistics and closely related questions of the structure of UMVUEs.In this paper a new method is developed that, in particular, proves that in the classical Nile problem no statistic subject to mild natural conditions is a UMVUE. The method almost solves an old problem of the existence of UMVUEs. The method is purely statistical (vs. analytical) and works for any family possessing an ancillary statistic. It complements an analytical method that uses only the first order ancillarity (and thus works when the existence of ancillary subalgebras is an open problem) and works for curved exponential families with polynomial constraints on the canonical parameters of which the Nile problem is a special case.

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Section: Statistical Methods and The Original Nile Problemmentioning

confidence: 99%

“…See also Hinkley [17] and Kariya [28] for related results. According to Nayak and Sinha [35], the problem of existence of UMVUEs is open.…”

Section: Problems Closely Related To the Nile Problemmentioning

confidence: 99%

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On the Nile problem by Sir Ronald Fisher

Kagan¹,

Malinovsky²

2013

Electron. J. Statist.

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“…We then show how the naive estimator can be improved using zero-estimators. Zero-estimators are introduced in the UMVUE literature [1,21,23], and are also used as a variance reduction technique in the Monte-Carlo simulation literature [3,12,19]. When the distribution of the covariates is known, an easy construction of many zero-estimators is feasible as shown in Section 4.…”

Section: Introductionmentioning

confidence: 99%

Improved estimators for semi-supervised high-dimensional regression model

Ilan

Azriel

Goldberg

2022

Electron. J. Statist.

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We study a high-dimensional linear regression model in a semisupervised setting, where for many observations only the vector of covariates X is given with no responses Y . We do not make any sparsity assumptions on the vector of coefficients, nor do we assume normality of the covariates. We aim at estimating the signal level, i.e., the amount of variation in the response that can be explained by the set of covariates. We propose an estimator, which is unbiased, consistent, and asymptotically normal. This estimator can be improved by adding zero-estimators arising from the unlabeled data. Adding zero-estimators does not affect the bias and potentially can reduce the variance. We further present an algorithm based on our approach that improves any given signal level estimator. Our theoretical results are demonstrated in a simulation study.

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“…The existence of UMVUEs in the Nile problem by R. Fisher (see Fisher (1936), Fisher (1973) and related problems attracted recently some attention (see Nayak and Sinha (2012)).…”

Section: Introductionmentioning

confidence: 99%

On the Structure of UMVUEs

Kagan

Malinovsky

2015

Sankhya A

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In all setups when the structure of UMVUEs is known, there exists a subalgebra U (MVE-algebra) of the basic σ-algebra such that all U-measurable statistics with finite second moments are UMVUEs. It is shown that MVE-algebras are, in a sense, similar to the subalgebras generated by complete sufficient statistics. Examples are given when these subalgebras differ, in these cases a new statistical structure arises.

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Some aspects of minimum variance unbiased estimation in presence of ancillary statistics

Cited by 3 publications

References 14 publications

On the Nile problem by Sir Ronald Fisher

On the Nile problem by Sir Ronald Fisher

Improved estimators for semi-supervised high-dimensional regression model

On the Structure of UMVUEs

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