On the Mean Square Error of Parameter Estimates for Some Biased Estimators

Lowerre, James M.

doi:10.1080/00401706.1974.10489217

Cited by 25 publications

(6 citation statements)

References 5 publications

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“…are less often used. [3][4][5][6][7][8][9][10][11][12][13][14][15][16] An attribute of these methods and other biased approaches to estimation of the regression vector is the reduction in bias at the expense of variance relative to the least squares model. Equations 1 and 2 can be rewritten in a variety of ways for each of the previously mentioned m ethods and for numerous other approaches depending on whether the eigenvector, PLS, a general orthogonal, or an oblique basis set is used.…”

Section: Introductionmentioning

confidence: 99%

Pareto Optimal Multivariate Calibration for Spectroscopic Data

Kalivas

Green

2001

Appl Spectrosc

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Multivariate calibration of spectra l data is considered with an emphasis on prediction. An abundance of methods are available to develop such calibration m odels. Using a harmonious approach with target vector optimization, the best calibration models are identi ed relative to the criteria used. Criteria utilized to determine the adequacy of models are minimization of the root mean square error of calibration (RMSEC) and the norm of the regressio n vector. Because of the simplicity of the optimization response surfaces, the method of sim plex was found to function much faster than generalized simulated annealing. Using a near infrared spectral example to demonstrate concepts, a family of models are established to be good, i.e., Pareto optimal models. For the data set investigated, it is found that the Pareto optimal models are essentially the same as models obtained by ridge regression and generalized ridge regression and are more harmonious than models obtained by principal component regressio n (PCR), partial least-squares (PLS), continuum regression, and cyclic subspace regression as the Pareto optimal models have smaller regressio n vector norms, RMSEC, and root mean square error of validation (RMSEV) values. The PLS models are found to be Pareto optimal relative to the PCR models. The paper presents an explanation of when the RR model will not be acceptable.

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Section: Introductionmentioning

confidence: 99%

Pareto Optimal Multivariate Calibration for Spectroscopic Data

Kalivas

Green

2001

Appl Spectrosc

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“…However, since s S 2 is a biased estimator of σ 2 (Hocking, 1976), statistical tests that use s S 2 can be misleading. When misspecifi ed models are analyzed (and the modeller believes that the model is misspecifi ed), it is common for the quality of parameter estimates (and model predictions) to be assessed using mean-squared-error (MSE) or mean-squarederror-matrix (MSEM) (Toutenburg and Trenkler, 1990;Price, 1982;Gunst and Mason, 1977;Lowerre, 1974), which account for both bias and variance. The MSEM and MSE for a parameter estimate β are defi ned as…”

Section: Misspecified Models-the General Casementioning

confidence: 99%

The Use of Simplified or Misspecified Models: Linear Case

Harris

McAuley

2007

Can J Chem Eng

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Les modèles simplifi és ont des propriétés intéressantes et présentent parfois de meilleures estimations de paramètres et prédictions de modèles, pour ce qui est de l'erreur quadratique moyenne, que les modèles plus élaborés, en particulier lorsque les données ne sont pas de type informatif. Nous présentons dans cet article un résumé d'un grand nombre de résultats quantitatifs et qualitatifs de la littérature scientifi que portant sur des modèles simplifi és ou mal spécifi és. En nous appuyant sur des intervalles de confi ance et des essais d'hypothèses, nous établissons une stratégie pratique afi n d'aider les concepteurs de modèles à déterminer s'ils doivent employer un modèle simplifi é et attirer leur attention sur la diffi culté de prendre une telle décision. Nous évaluons également plusieurs méthodes d'inférence statistique pour des modèles simplifi és ou mal spécifi és.

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“…Developed by Hoerl and Kennard [7, 8], and tested and extended by others [1, 4, 12, 14, 15], ridge regression is a multivariate technique which uses a modification of the OLS formula designed to minimize the variance in γ ( VAR γ ) by accepting a small amount of bias into the estimate.…”

Section: Ridge Regressionmentioning

confidence: 99%

Ridge Regression and the Multicollinearity Problem in Financial Research: A Case Study

Marquette

Johnson

1980

J of Financial Research

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The explanation and prediction of security returns and their relation to risk has received a great deal of attention in the financial literature. Both intuitive and theoretical models have been developed in which either return or risk is expressed as a linear function of either one or several macroeconomic, market, or firm-related variables. Studies attempting to explore these relationships, however, have been plagued by the interdependent nature of corporate financial variables. When using classical multiple regression analysis (OLS), these interdependencies may result in an ill-conditioned, multicollinear matrix. Inability or failure to compensate leaves the researcher to cope with the various symptoms of multicollinearity including overstated regression coefficients, incorrect signs, and highly unstable predictive equations.Until recently, there has been little the researcher could do when faced with multicollinear data. Elimination of interdependent variables may well reduce the explanatory power of the model to the point of sterility; collecting more data is frequently impossible or implausible; and clustering and factoring techniques often leave variables which, if orthogonal, are also uninterpretable.Recently, however, two new techniques have been developed to deal with multicollinearity without major departures from the OLS model. They are the generalized inverse [13] and ridge regression. Both are modifications of classical OLS. Both help to eliminate the earlier mentioned problems inherent in working with OLS and an ill-conditioned matrix.The purpose of this paper is to present ridge regression as an alternative technique to OLS for the empirical testing of financial models in which a high degree of multicollinearity exists, particularly those designed to predict the market risk/return relationship. While the use of ridge regression should improve the predictive power of any financial model possessing a high degree of multicollinearity, two groups of financial models in particular have been well noted for the statistical problems of interdependent predictor variables.The first group consists of the multi-index models developed as extensions to the single index market model, and the second group consists of those financial models specifying systematic risk or return as a function of corporate financial and/or operating variables. In many of these studies the authors have been hampered by the multicollinear nature of the variables under examination [2,3,5,6, 11,16,17,19]. In some cases the authors have resorted to homogeneous industry samples or to factor analysis in order to avoid problems associated with interdependent predictor variables. While these studies have dealt with the statistical problems of multicollinearity, in so doing, the researchers have often been forced to discard information or to deviate from a cleanly interpretable research design.

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On the Mean Square Error of Parameter Estimates for Some Biased Estimators

Cited by 25 publications

References 5 publications

Pareto Optimal Multivariate Calibration for Spectroscopic Data

Pareto Optimal Multivariate Calibration for Spectroscopic Data

The Use of Simplified or Misspecified Models: Linear Case

Ridge Regression and the Multicollinearity Problem in Financial Research: A Case Study

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