## The Annals of Statistics

### Minimum Chi-Square, not Maximum Likelihood!

Joseph Berkson

#### Abstract

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

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

Dates
First available: 12 April 2007

http://projecteuclid.org/euclid.aos/1176345003

JSTOR

Digital Object Identifier
doi:10.1214/aos/1176345003

Mathematical Reviews number (MathSciNet)
MR568715

Zentralblatt MATH identifier
0456.62023

Subjects
Primary: 62F10: Point estimation
Secondary: 62F20

#### Citation

Berkson, Joseph. Minimum Chi-Square, not Maximum Likelihood!. The Annals of Statistics 8 (1980), no. 3, 457--487. doi:10.1214/aos/1176345003. http://projecteuclid.org/euclid.aos/1176345003.