Bounding maximum likelihood estimates based on incomplete ordered data

In general, the maximum likelihood estimators (MLEs) of the parameters of one- and two-parameter exponential models based on incomplete ordered data do not admit closed form expressions. Instead of obtaining linear approximations to the MLEs, as is common in the statistical literature, explicit and precise non-linear under- and over-estimates are provided. The results derived can also be applied to some other models, as Pareto, Weibull with constant shape, Burr Types X and XII, and power-function distributions. The proposed lower and upper bounds are usually superior to approximate MLEs, and also can serve as starting points for iterative interpolation methods such as regula falsi. Due to the sharpness of the bounds, midpoints are excellent approximations to MLEs in most practical cases. As an additional advantage, the estimation errors of the midpoints can be accurately bounded. An illustrative example and some comments about linear estimation, asymptotics and expected Fisher information are also included.