Results being based on the covariance matrix and its cofactors, we give explicit expressions for the asymptotic bias, 2nd order variances, skewness to order  and asymptotic kurtosis to order 1/N, N being the sample size. Except for the simultaneous estimation of a, b, c, the expressions for these asymptotic moments and moment ratios are simple in form involving gamma and Riemann Zeta functions. They provide a new basic supplement to our knowledge of maximum likelihood estimator moments.

A surprising discovery is the part played by the location parameter whenever it has to be estimated. For the three parameter estimation case it is already known that asymptotic covariance only exist if c > 2.  It turns out that the asymptotic skewness only exist if c > 3 and the asymptotic kurtosis only exist if c > 4. This applies to the asymptotic distribution of and  The source of this characteristic is the singularity appearing in the expectation of logarithmic derivatives. When less than 3 parameters are to be estimated the problem arises whenever  intrudes.

For the 3 parameter case, a new expression is developed for the asymptotic variance of  Lastly, wherever possible simulation studies are invoked for verification purposes.