We consider a lower bound for the variance of a regular estimating function for $\theta_1$, the parameter of interest, when there are $m(\ge 1)$ nuisance parameters $\theta_{2i}$, $i=1,\ldots,m$. Godambe's (1960) lower bound of the variance of a regular unbiased estimating function (with no nuisance parameter present) is valid under some regularity conditions. Considering Chapman-Robbins-Kiefer inequality we obtain this note another lower of variance of an estimating function under condition which do not require Godambe's regularity conditions. Finally we indicate two extensions of the general form of an optimal estimating function proposed by Godambe and Thompson (1976) in the presence of parameter.