On the Problem of Calibration

Shukla, G. K.

doi:10.1080/00401706.1972.10488944

Cited by 75 publications

(32 citation statements)

References 9 publications

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“…5. This huge credible interval is in agreement with the frequentist mean square error of 4.9 obtained with the expansion of Shukla (1972). 3, but now with the NAO as the independent variable.…”

Section: A the Nao-nam Examplesupporting

confidence: 88%

Straight Line Fitting and Predictions: On a Marginal Likelihood Approach to Linear Regression and Errors-In-Variables Models

Christiansen

2014

Journal of Climate

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Even in the simple case of univariate linear regression and prediction there are important choices to be made regarding the origins of the noise terms and regarding which of the two variables under consideration that should be treated as the independent variable. These decisions are often not easy to make but they may have a considerable impact on the results. A unified probabilistic (i.e., Bayesian with flat priors) treatment of univariate linear regression and prediction is given by taking, as starting point, the general errors-in-variables model. Other versions of linear regression can be obtained as limits of this model. The likelihood of the model parameters and predictands of the general errors-in-variables model is derived by marginalizing over the nuisance parameters. The resulting likelihood is relatively simple and easy to analyze and calculate. The wellknown unidentifiability of the errors-in-variables model is manifested as the absence of a well-defined maximum in the likelihood. However, this does not mean that probabilistic inference cannot be made; the marginal likelihoods of model parameters and the predictands have, in general, well-defined maxima. A probabilistic version of classical calibration is also included and it is shown how it is related to the errors-invariables model. The results are illustrated by an example from the coupling between the lower stratosphere and the troposphere in the Northern Hemisphere winter.

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Section: A the Nao-nam Examplesupporting

confidence: 88%

Straight Line Fitting and Predictions: On a Marginal Likelihood Approach to Linear Regression and Errors-In-Variables Models

Christiansen

2014

Journal of Climate

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“…The variance of the classical estimator is infinite, but Shukla (1972) obtained expressions for the expectation and the mean squared error of (3.1), conditional on the event that I /BI > 0, to terms of O(n-~) assuming that F is normal. Shukla's approach can be extended to the case in which F is a more general distribution with the following specifications: (3.5)…”

Section: The Case When a /F And A2 Are Unknownmentioning

confidence: 99%

“…For normal F, Shukla (1972) obtained expressions for the expectation and the mean squared error of (3.5) to terms of O(n-1). Again, Shukla's result for the inverse estimator can be extended to a more general distribution of Y as given in ( B2=43n -l 3A -3Sxl, C2 0 -2-2 + n-1 + A20 -2(2 -2 + 6)…”

Section: The Case When a /F And A2 Are Unknownmentioning

confidence: 99%

An Analysis of the Linear-Calibration Controversy from the Perspective of Compound Estimation

Lwin¹,

Maritz²

1982

Technometrics

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“…In practice, of course, we must use

Ẑ

defined analogously to , and

{\overset{ξ}{true}}_{c}

as defined in ; and then, as with the classical estimator in the context of simple linear regression, division by

{\overset{β}{true}}_{1}

results in infinite MSE for both estimators. However, in typical applications, there is a minimal probability of

{\overset{β}{true}}_{1}

being close to zero (note the confidence interval in Table ), so as with the classical estimator , one can consider asymptotic MSE, conditional on

MathClass-rel∣ {\overset{β}{true}}_{1} MathClass-rel∣ MathClass-rel> 0

. It is difficult to obtain a usable analytic approximation to the asymptotic

MSE ({\overset{ξ}{true}}_{c} MathClass-punc, ξ)

; moreover, such an approximation, using the delta method, would require assuming not only that the estimates

\overset{β}{true} MathClass-punc, {\overset{σ}{true}}^{2}

and

\overset{bold-italicΦ}{true}

are close to the true parameter values in – but also that ε 0 is close to zero in .…”

Section: Numerical Performancementioning

confidence: 99%

Shrinkage estimation and the use of additional information when calibrating in the presence of random effects

Oman

2013

Statistics in Medicine

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Let x denote a precise measurement of a quantity and Y an inexact measurement, which is, however, less expensive or more easily obtained than x. We have available a calibration set comprising clustered sets of (x,Y) observations, obtained from different sampling units. At the prediction step, we will only observe Y for a new unit, and we wish to estimate the corresponding unknown x, which we denote by ξ. This problem has been treated under the assumption that x and Y are linearly related. Here, we expand on those results in three directions: First, we show that if we center ξ about a known value c, for example, the mean x-value of the calibration set, then the proposed estimator now shrinks to c. Second, we examine in detail the performance of the estimator, which was proposed when one or more (x,Y) observations can be obtained for the new subject. Third, we compare the Fieller-like confidence intervals, previously proposed, with t-like intervals based on asymptotic moments of the point estimate. We illustrate and evaluate our procedures in the context of a data set of true bladder-volumes (x) and ultrasound measurements (Y).

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On the Problem of Calibration

Cited by 75 publications

References 9 publications

Straight Line Fitting and Predictions: On a Marginal Likelihood Approach to Linear Regression and Errors-In-Variables Models

Straight Line Fitting and Predictions: On a Marginal Likelihood Approach to Linear Regression and Errors-In-Variables Models

An Analysis of the Linear-Calibration Controversy from the Perspective of Compound Estimation

Shrinkage estimation and the use of additional information when calibrating in the presence of random effects

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