On a property of distributions admitting sufficient statistics

A property of distributions admitting sufficient statistics is obtained, connecting the likelihood function of a sample of n observations, the maximum likelihood estimates of the parameters and the information matrix. A geometric meaning of the property is given. The property is used in simplifying the calculations of the variances and covariances of the maximum likelihood estimates in large samples. Finally, it is shown in virtue of the property that the likelihood equations have a unique solution for every sample of any size, and that the solution does make the likelihood function a maximum