What Did Fisher Mean by An Estimate?

TL;DR

This paper clarifies Fisher's original concept of estimation as constructing likelihood intervals, and proposes a new method for calculating profile likelihood-based confidence intervals aligned with Fisher's ideas.

Contribution

It reinterprets Fisher's notion of estimation, emphasizing likelihood intervals, and introduces a novel approach for computing profile likelihood confidence intervals for complex models.

Findings

Likelihood intervals are central to Fisher's estimation concept.

Profile likelihood-based confidence intervals can be computed for general models.

The new method aligns confidence interval construction with Fisher's original ideas.

Abstract

Fisher's Method of Maximum Likelihood is shown to be a procedure for the construction of likelihood intervals or regions, instead of a procedure of point estimation. Based on Fisher's articles and books it is justified that by estimation Fisher meant the construction of likelihood intervals or regions from appropriate likelihood function and that an estimate is a statistic, that is, a function from a sample space to a parameter space such that the likelihood function obtained from the sampling distribution of the statistic at the observed value of the statistic is used to construct likelihood intervals or regions. Thus Problem of Estimation is how to choose the 'best' estimate. Fisher's solution for the problem of estimation is Maximum Likelihood Estimate (MLE). Fisher's Theory of Statistical Estimation is a chain of ideas used to justify MLE as the solution of the problem of estimation. The construction of confidence intervals by the delta method from the asymptotic normal distribution of MLE is based on Fisher's ideas, but is against his 'logic of statistical inference'. Instead the construction of confidence intervals from the profile likelihood function of a given interest function of the parameter vector is considered as a solution more in line with Fisher's 'ideology'. A new method of calculation of profile likelihood-based confidence intervals for general smooth interest functions in general statistical models is considered.