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Mary Immaculate College Research Seminar September 3 rd 2013 Anomalies of the Maximum Likelihood Estimator Diarmuid O’Driscoll Mary Immaculate College Limerick Donald E Ramirez University of Virginia Charlottesville

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Least Squares Regression Line Least Squares Regression Line in the Measurement Error Model has wide interest with many applications Studied in depth by Carroll et al. (2006) and Fuller (1987) OLS(Y|X) assumes that measurement of X is error free

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Minimizing Oblique Errors

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Standardized Weighted Model

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Quartic

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Measurement Error Model

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Specific values of lambda

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Bias

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The Maximum Likelihood Estimator

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Monte Carlo Simulation

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−6.0−0.0298−0.0296− −4.0−0.0201−0.0198− −2.0− − −2.0−0.0089−0.0099− −1.0−0.0048−0.0050−

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Second and Fourth Moments

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The Frisch hyperbola of Van Montfort (1989) is and the fourth order moment equation Moment Estimating Equations

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True Ratio:Solution:

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True Ratio: Solution:

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Two new oblique estimators

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Monte Carlo Simulation

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Bibliography Adcock, R. J. (1878). 'A problem in least-squares', The Analyst, 5: Al-Nasser A. (2012). 'On using the maximum entropy median for fitting the un- replicated functional model between the unemployment rate and the human development index in the Arab states', J. Appl. Sci., 12: 326–335. Carroll, R. J., Ruppert, D., Stefanski, L. A., Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models - A Modern Perspective, Boca Raton: Chapman & Hall/CRC, Second Edition. Deming, W. E. (1943). Statistical Adjustment of Data, New York: Wiley. Fuller, W.A. (1987). Measurement Error Models, New York: Wiley. Gillard, J., Iles T. (2009). 'Methods of fitting straight lines where both variables are subject to measurement error', Current Clinical Pharmacology, 4: O'Driscoll, D., Ramirez, D. (2011). 'Geometric View of Measurement Errors', Communications in Statistics-Simulation and Computation, 40: Van Montfort, K., Mooijaart, A., and de Leeuw, J. (1987). 'Regression with errors in variables', Statist Neerlandica, 41: Yang L. (1999). 'Recent advances on determining the number of real roots of parametric polynomials', J. Symbolic Computation, 28:

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