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 ϕ = ϕ(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.