AbstractIf Y1 ≤ … ≤ Yn are ordered observations from a population with cumulative distribution function \documentclass{article}\pagestyle{empty}\begin{document}${\rm G}\left( {{\textstyle{{{\rm X - B}} \over {\rm C}}}} \right)$\end{document}, probability density function \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm (1/C)g}\left( {{\textstyle{{{\rm X - B}} \over {\rm C}}}} \right) $\end{document}, where B and C are unknown parameters with C > 0, and the function G(x) is known, it is shown that under mild restrictions on G(x), \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{{\rm n + 1}}}{{{\rm n - 1}}}\sum\limits_{{\rm j = 1}}^{{\rm n - 1}} {{\rm g[G}^{{\rm - 1}} {\rm (j/n)](Y}_{{\rm j + 1}} {\rm - Y}_{\rm j} {\rm)}} $\end{document} is a consistent estimate of C. In certain important cases, this estimate has a structure similar to that of estimates known to be optimal.