The Dual of the Maximum Likelihood

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DOI: 10.4236/ojs.2016.61016    1,579 Downloads   1,833 Views  
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The Maximum Likelihood method estimates the parameter values of a statistical model that maximizes the corresponding likelihood function, given the sample information. This is the primal approach that, in this paper, is presented as a mathematical programming specification whose solution requires the formulation of a Lagrange problem. A result of this setup is that the Lagrange multipliers associated with the linear statistical model (where sample observations are regarded as a set of constraints) are equal to the vector of residuals scaled by the variance of those residuals. The novel contribution of this paper consists in deriving the dual model of the Maximum Likelihood method under normality assumptions. This model minimizes a function of the variance of the error terms subject to orthogonality conditions between the model residuals and the space of explanatory variables. An intuitive interpretation of the dual problem appeals to basic elements of information theory and an economic interpretation of Lagrange multipliers to establish that the dual maximizes the net value of the sample information. This paper presents the dual ML model for a single regression and provides a numerical example of how to obtain maximum likelihood estimates of the parameters of a linear statistical model using the dual specification.

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Paris, Q. (2016) The Dual of the Maximum Likelihood. Open Journal of Statistics, 6, 186-193. doi: 10.4236/ojs.2016.61016.


[1] Kmenta, J. (2011) Elements of Econometrics. 2nd Edition, The University of Michigan Press, Ann Arbor.
[2] Stock, J.H. and Watson, M.W. (2011) Introduction to Econometrics. 3rd Edition, Addison-Wesley, Boston.
[3] Brooke, A., Kendrick, D. and Meeraus, A. (1988) GAMS—A User’s Guide. The Scientific Press, Redwood City.
[4] Paris, Q. (2015) The Dual of the Least-Squares Method. Open Journal of Statistics, 5, 658-664.
[5] Heady, E.O., Pesek, J.T. and Brown, W.G. (1955) Corn Response Surfaces and Economic Optima in Fertilizer Use. Iowa State Experimental Station, Bulletin 424.

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