Maximum Likelihood Estimation of Means and Variances from Normal Populations Under Simultaneous Order Restrictions

For k normal populations with unknown means [mu]i and unknown variances [sigma]2i, i = 1, ..., k, assume that there are some order restrictions among the means and variances, respectively, for example, simple order restrictions: [mu]1 <= [mu]2 <= ... <= [mu]k and [sigma]21 >= [sigma]22 >= ... >= [sigma]2k > 0. Some properties of maximum likelihood estimation of [mu]is and [sigma]2i are discussed and an algorithm of obtaining the maximum likelihood estimators under the order restrictions is proposed.