The Annals of Statistics

Minimum Chi-Square, not Maximum Likelihood!

Joseph Berkson

Full-text: Open access


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.

Article information

Ann. Statist., Volume 8, Number 3 (1980), 457-487.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F10: Point estimation
Secondary: 62F20

Estimation criteria of estimate maximum likelihood minimum chi-square efficiency second order efficiency


Berkson, Joseph. Minimum Chi-Square, not Maximum Likelihood!. Ann. Statist. 8 (1980), no. 3, 457--487. doi:10.1214/aos/1176345003.

Export citation