For a one parameter parametric family a study is made of the total variation distance, D, between the unknown distribution and the estimated distribution , where is the distance, between the unknown distribution , and the estimated distribution where is the maximum likelihood estimator of based on a sample from . Knowledge of the distribution of D yields simultaneous confidence intervals for where is the class of Borel sets in . In the case of a normal distribution with known variance the exact distribution is obtained. More generally the asymptotic distribution of is derived and explicitly computed for several families of interest. For an m vector, , an asymptotically conservative set of confidence intervals for is constructed.
