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

For k normal populations with unknown means μi and unknown variances σ2i, i = 1, ..., k, assume that there are some order restrictions among the means and variances, respectively, for example, simple order restrictions: μ1 ≤ μ2 ≤ ... ≤ μk and σ21 ≥ σ22 ≥ ... ≥ σ2k > 0. Some properties of maximum likelihood estimation of μis and σ2i are discussed and an algorithm of obtaining the maximum likelihood estimators under the order restrictions is proposed.