Problems with maximum likelihood estimation and the 3 parameter gamma distribution

The three parameters involved are scale a , shape rho, and location s . Maximum likelihood estimators are ((a) over cap, (rho) over cap, (s) over cap) . Using recent work on the second order variances, skewness, and kurtosis we establish the facts, that if the location parameter s is to be estimated, then the asymptotic variances only exist if rho>2, asymptotic skewness only exists if rho>3, and 2nd order variances and third order fourth central moments only exist if rho>4. The result of these limitations is that in general very large sample sizes may be needed to avoid inference problems. We also include new continued fractions for the asymptotic covariances of the maximum likelihood estimators considered.