On the Bias and Mean-square Error of Order-restricted Maximum Likelihood Estimators

An order-restricted (OR) statistical model can be expressed in the general form {Pθ | θ ∈ C}, where C is a convex cone in R. In general, no unbiased estimator exists for θ. In particular, the OR maximum likelihood estimator (ORMLE) is biased, although its aggregate mean square error is usually less than that of the unrestricted MLE (URMLE). Nonetheless, the bias and mean-square error (MSE) of a single component or single linear contrast of the ORMLE can exceed those of the corresponding component or contrast of the URMLE by amounts that approach infinity as the dimension increases. This phenomenon is examined in detail for three examples: the orthant cone, the tree-order cone, and the simple-order cone. The geometric features of the cone that determine the growth rate of the bias and MSE are studied, and bias-reducing adjustments for certain components or contrasts of the ORMLE are suggested for the orthant and tree-order models.