Combining Unbiased Estimators

Graybill, Franklin A.; Deal, Roy B.

doi:10.2307/2527652

Cited by 265 publications

(115 citation statements)

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“…Within at least the CCQM, except when the KCRV has been assigned using the Graybill-Deal estimator [21,22], the q(x,V KC ) term has generally been ignoredeffectively asserting that q(x,V KC ) = 0. Since the MRA requires that uncertainties are to be specified at the 95 % level of confidence, standard uncertainties must usually be estimated from reported expanded uncertainties…”

Section: Resultsmentioning

confidence: 99%

“…Key comparison reference value, V KC While many location estimators have been proposed for evaluating a KCRV and recent guidance provided for choosing and calculating ones appropriate to particular circumstances [23], all of the KCs considered here have used either the median when there was significant between-result variance, s 2 b , or the Graybill-Deal weighted mean [21], x GD , when s 2 b was considered insignificant. The x GD is defined as…”

Section: Revisiting the Estimation Of Degrees Of Equivalencementioning

confidence: 99%

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“Degrees of equivalence” for chemical measurement capabilities: primary pH

Duewer

Pratt

Cherdchu³

et al. 2014

Accred Qual Assur

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The key comparison (KC) studies of the Consultative Committee for Amount of Substance-Metrology in Chemistry help ensure the reliability of chemical and biochemical measurements relevant to international trade and environmental-, health-, and safety-related decision making. The traditional final evaluation of each measurement result reported by a KC participant is a ''degree of equivalence'' (DEq) that quantitatively specifies how consistent each individual result is relative to a reference value. Recognizing the impossibility of conducting separate KCs for all important analytes in all important sample matrices at all important analyte levels, emphasis is now shifting to documenting broadly applicable critical or ''core'' measurement competencies elicited through a series of studies. To better accomplish the necessary synthesis of results, data analysis and display tools must be developed for objectively and quantitatively combining individual DEqs. The information detailed in the 11 KCs of primary method pH measurements publically available as of 2013 provides an excellent ''best case'' prototype for such analysis. We here propose tools that enable documenting the expected primary pH measurement performance of individual participants between pH 1 and pH 11 and from 15°C to 37°C. These tools may prove useful for other areas where the uncertainty of measurement is a predictable function of the measured quantity, such as the stable gases. That results for relatively simple measurement processes can be combined using relatively simple analysis and display methods does not ensure that similarly meaningful summaries can be devised for less well understood and controlled systems, but it provides the incentive to attempt to do so.

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Section: Resultsmentioning

confidence: 99%

Section: Revisiting the Estimation Of Degrees Of Equivalencementioning

confidence: 99%

“Degrees of equivalence” for chemical measurement capabilities: primary pH

Duewer

Pratt

Cherdchu³

et al. 2014

Accred Qual Assur

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“…Now, suppose that the covariance matrices are unknown and unequal, then motivated by the idea of Graybill and Deal (1959) and also Robertson et al (1988) in univariate case, we estimate the unknown covariance matrices, Σ i by S i , i = 1, 2. Therefore, we havê…”

Section: The Main Theoremsmentioning

confidence: 99%

On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Bazyari¹

2015

JSTA

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Problems concerning estimation of parameters and determination the statistic, when it is known a priori that some of these parameters are subject to certain order restrictions, are of considerable interest. In the present paper, we consider the estimators of the monotonic mean vectors for two dimensional normal distributions and compare those with the unrestricted maximum likelihood estimators under two different cases. One case is that covariance matrices are known, the other one is that covariance matrices are completely unknown and unequal. We show that when the covariance matrices are known, under the squared error loss function which is similar to the mahalanobis distance, the obtained multivariate isotonic regression estimators, motivated by estimators given in Robertson et al. (1988), which are the estimators given by Sasabuchi et al. (1983) and Sasabuchi et al. (1992), have the smaller risk than the unrestricted maximum likelihood estimators uniformly, but when the covariance matrices are unknown and unequal, the estimators have the smaller risk than the unrestricted maximum likelihood estimators only over some special sets which are defined on the covariance matrices. To illustrate the results two numerical examples are presented.

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“…The common mean problem has a long history of research since the pioneering work by Graybill and Deal (1959). It has real-life applications in wide areas, for example in clinical trials (Kelleher, 1996), in soil engineering (Zacks, 1966(Zacks, , 1970, in a balanced incomplete block design (Montgomery, 1997;Pal et al, 2007), and in meta analysis (Chang and Pal, 2008).…”

Section: Introductionmentioning

confidence: 99%

Influence Analysis of the Common Mean Problem

Kim

2013

Communications for Statistical Applications and Methods

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Two influence diagnostic methods for the common mean model are proposed. First, an investigation of the influence of observations according to minor perturbations of the common mean model is made by adapting the local influence method which is based on the likelihood displacement. It is well known that the maximum likelihood estimates are in general sensitive to influential observations. Case-deletions can be a candidate for detecting influential observations. However, the maximum likelihood estimators are iteratively computed and therefore case-deletions involve an enormous amount of computations. An approximation by Newton's method to the maximum likelihood estimator obtained after a single observation was deleted can reduce much of computational burden, which will be treated in this work. A numerical example is given for illustration and it shows that the proposed diagnostic methods can be useful tools.

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Combining Unbiased Estimators

Cited by 265 publications

References 1 publication

“Degrees of equivalence” for chemical measurement capabilities: primary pH

“Degrees of equivalence” for chemical measurement capabilities: primary pH

On the Properties of Estimates of Monotonic Mean Vectors for Multivariate Normal Distributions

Influence Analysis of the Common Mean Problem

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