n where m8 = E xj/n, ,Is = 6'ms and q is a positive integer. The moments s are functions of the j=l h parameters (01, 02.O h)_ 0, and it is assumed that the second derivatives with respect to these parameters exist in some 0-interval. The range of the variate x is assumed to be independent of 0. The equations D.(O) = 0 (a = 1, 2, ..., h) are a generalized form of the moment estimators discussed by one of us previously (Shenton, 1958, 1959). In general the properties of moment estimators are expected to be similar to those based on the first few moments (means, covariances, etc.) when q is small, and similar to maximum likelihood estimators when q is large. Our object here is to obtain expressions for the asymptotic biases of moment estimators, and also incidentally expressions for the moment estimator covariances. We then let q -? oo and so find expressions for the biases of maximum likelihood estimators. As illustrations we derive the biases for the parameters in normal and bivariate normal distributions, and as an application we give an assessment of the biases for the negative binomial distribution. 2. EXPANSION FOR A ROOT OF Da(O) = 0 Introducing the cofactors Xsa (S = 0, 1, ..., q) of the elements in the first row of Da(O) we have (using the summation convention, with the understanding that the range of subscripts and superscripts is 0 to q for Roman characters and from 1 to h for Greek characters)