On Estimating Linear Relationships When Both Variables Are Subject to Heteroscedastic Measurement Errors

Cheng, Chi-Lun; Riu, Jordi

doi:10.1198/004017006000000237

Cited by 65 publications

(81 citation statements)

References 19 publications

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“…Then it can be shown that the MLEα andβ given by (4) are asymptotically efficient in the sense of Hajek bounds [18]. They are also optimal in other ways; see Gleser [17] and a survey [11]. We note that the so-called adjusted least squares estimators of α and β also have infinite moments [10], and they are efficient in the sense of Hajek bounds, too [26].…”

Section: Curve Fitting Methods In Errors-in-variables Modelsmentioning

confidence: 99%

Statistical analysis of curve fitting methods in errors-in-variables models

Al-Sharadqah

Chernov

2012

Theor. Probability and Math. Statist.

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Abstract. Regression models in which all variables are subject to errors are known as errors-in-variables (EIV) models. The respective parameter estimates have many unusual properties: their exact distributions are very hard to determine, and their absolute moments are often infinite (so that their mean and variance do not exist). In our paper, Error analysis for circle fitting algorithms, Electr. J. Stat. 3 (2009), 886-911, we developed an unconventional statistical analysis that allowed us to effectively assess EIV parameter estimates and design new methods with superior characteristics. In this paper we validate our approach in a series of numerical tests. We also prove that in the case of fitting circles, the estimates of the parameters are absolutely continuous (have densities).

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Section: Curve Fitting Methods In Errors-in-variables Modelsmentioning

confidence: 99%

Statistical analysis of curve fitting methods in errors-in-variables models

Al-Sharadqah

Chernov

2012

Theor. Probability and Math. Statist.

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“…[26] Despite its apparent simplicity, the straight line model ''when both variables are subject to error'' hides complex difficulties and has been the focus of a large number of publications from various research areas: statistics [Lindley and El-Sayyad, 1968;Kendall and Stuart, 1983;Fuller, 1987;Cheng and Ness, 1994], econometrics [Zellner, 1971;Erickson, 1989], physics [York, 1966;Reed, 1989;Gull, 1989], and image reconstruction [Werman and Keren, 2001]. In the following, we will review the standard least squares approach (also called ordinary least squares) and its inherent bias in presence of input errors.…”

Section: First Application: the Straight Line Modelmentioning

confidence: 99%

A Bayesian perspective on input uncertainty in model calibration: Application to hydrological model “abc”

Huard

Mailhot

2006

Water Resources Research

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[1] The impact of input errors in the calibration of watershed models is a recurrent theme in the water science literature. It is now acknowledged that hydrological models are sensitive to errors in the measures of precipitation and that those errors bias the model parameters estimated via the standard least squares (SLS) approach. This paper presents a Bayesian uncertainty framework allowing one to account for input, output, and structural (model) uncertainties in the calibration of a model. Using this framework, we study the impact of input uncertainty on the parameters of the hydrological model ''abc.'' Mostly of academic interest, the ''abc'' model has a response linear to its input, allowing the closed form integration of nuisance variables under proper assumptions. Using those analytical solutions to compute the posterior density of the model parameters, some interesting observations can be made about their sensitivity to input errors. We provide an explanation for the bias identified in the SLS approach and show that in the input error context the prior on the input ''true'' value has a significant influence on the parameters' posterior density. Overall, the parameters obtained from the Bayesian method are more accurate, and the uncertainty over them is more realistic than with SLS. This method, however, is specific to linear models, while most hydrological models display strong nonlinearities. Further research is thus needed to demonstrate the applicability of the uncertainty framework to commonly used hydrological models.

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“…De nombreux articles existent sur le sujet : ratio des variances constant, connu ou inconnu, droite de régression orthogonale. Dans certains cas, ces méthodes donnent des résultats équivalents [8].…”

Section: Méthodes D'estimationunclassified

Estimer la droite d’étalonnage avec les moindres carrés généralisés et évaluer le résultat de mesure

Yardin¹

2013

R. franç. métrol.

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RésuméLa méthode des moindres carrés généralisés (GLS) estime la droite d'étalonnage en tenant compte de l'incertitude associée aux variables, de l'hétéroscédasticité et des corrélations. Mais l'efficacité des GLS requiert une bonne connaissance de la méthode elle-même et du processus d'étalonnage. Cet article précise les caractéristiques de la fonction d'étalonnage et introduit le « modèle à erreurs sur les variables » plus représentatif du contexte. La méthode GLS est présentée selon diffé-rents scénarios d'incertitude en considérant les trois phases de l'estimation : modélisation, évaluation et validation. La propagation de l'incertitude des variables aux coefficients de la droite est discutée. Puis, le résultat de mesure est calculé avec la droite estimée, en mode direct et en mode inverse. Son incertitude est donnée en distinguant la valeur vraie ou moyenne sans incertitude et une valeur mesurée ou individuelle avec incertitude. Deux exemples sont analysés. Les incertitudes des variables influencent peu la valeur des coefficients de la droite et du résultat. En revanche, les incertitudes associées sont très affectées. MOTS CLÉS : MOINDRES CARRÉS GÉNÉRALISÉS, MODÈLE À ERREURS SUR LES VARIABLES, DROITE D'ÉTALONNAGE, DROITE AVEC INCERTITUDE SUR LES DEUX VARIABLES. Abstract The Generalized Least Squares (GLS) method estimates the straight-line calibration function by taking into account uncertainties in variables, heteroscedasticity and correlations. However, the efficiency of the method depends on a good knowledge of the method itself and of the calibration process. This paper details the characteristics of the calibration function and introduces the "model with errors in vari-

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On Estimating Linear Relationships When Both Variables Are Subject to Heteroscedastic Measurement Errors

Cited by 65 publications

References 19 publications

Statistical analysis of curve fitting methods in errors-in-variables models

Statistical analysis of curve fitting methods in errors-in-variables models

A Bayesian perspective on input uncertainty in model calibration: Application to hydrological model “abc”

Estimer la droite d’étalonnage avec les moindres carrés généralisés et évaluer le résultat de mesure

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