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May, 1980 Minimum Chi-Square, not Maximum Likelihood!
Joseph Berkson
Ann. Statist. 8(3): 457-487 (May, 1980). DOI: 10.1214/aos/1176345003


The sovereignty of MLE is questioned. Minimum $\chi^2_\lambda$ yields the same estimating equations as MLE. For many cases, as illustrated in presented examples, and further algorithmic exploration in progress may show that for all cases, minimum $\chi^2_\lambda$ estimates are available. In this sense minimum $\chi^2$ is the basic principle of estimation. The criterion of asymptotic sufficiency which has been called "second order efficiency" is rejected as a criterion of goodness of estimate as against some loss function such as the mean squared error. The relation between MLE and sufficiency is not assured, as illustrated in an example in which MLE yields $\infty$ as estimate with samples that have different values of the sufficient statistic. Other examples are cited in which minimal sufficient statistics exist but where the MLE is not sufficient. The view is advanced that statistics is a science, not mathematics or philosophy (inference) and as such requires that any claimed attributes of the MLE must be testable by a Monte Carlo experiment.


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Joseph Berkson. "Minimum Chi-Square, not Maximum Likelihood!." Ann. Statist. 8 (3) 457 - 487, May, 1980.


Published: May, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0456.62023
MathSciNet: MR568715
Digital Object Identifier: 10.1214/aos/1176345003

Primary: 62F10
Secondary: 62F20

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 3 • May, 1980
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