The interpretation of maximum-likelihood estimation

Maximum-likelihood estimation is interpreted as a procedure for generating approximate pivotal quantities, that is, functions u(X;θ) of the data X and parameter θ that have distributions not involving θ. Further, these pivotals should be efficient in the sense of reproducing approximately the likelihood function of θ based on X, and they should be approximately linear in θ. To this end the effect of replacing θ by a parameter ? = ?(θ) is examined. The relationship of maximum-likelihood estimation interpreted in this way to conditional inference is discussed. Examples illustrating this use of maximum-likelihood estimation on small samples are given.