On Small-Sample Estimation

This paper discusses some of the concepts underlying small sample estimation and reexamines, in particular, the current notions on "unbiased" estimation. Alternatives to the usual unbiased property are examined with respect to invariance under simultaneous one-to-one transformation of parameter and estimate; one of these alternatives, closely related to the maximum likelihood method, seems to be new. The property of being unbiased in the likelihood sense is essentially equivalent to the statement that the estimate is a maximum likelihood estimate based on some distribution derived by integration from the original sampling distribution, by virtue of a "hereditary" property of maximum likelihood estimation. An exposition of maximum likelihood estimation is given in terms of optimum pairwise selection with equal weights, providing a type of rationale for small sample estimation by maximum likelihood.