Inferences Under a Stochastic Ordering Constraint

Constrained estimation and the theorem of Kuhn-Tucker

“…Varying the value of x we derive estimates for the distribution functions over their entire range. Pursuing the mathematics, we recover the estimates derived initially by Hogg [10] and discussed in depth by El Barmi and Mukerjee [5] and the references therein. We note that this estimator is not the nonparametric maximum likelihood estimator derived by Brunk et al [3].…”

Order restricted inference for comparing the cumulative incidence of a competing risk over several populations

“…The extension of point-wise convergence in distribution to weak convergence using assumption (2.10) is straightforward (see proof of Theorem 4 in EBM [3]). …”

“…These results can be substantially sharpened when k = 2. We refer the reader to Section 4.2 in EBM [3].…”

“…, F k1 that satisfy (1.3), prove their consistency, derive their asymptotic properties, show the improvements over the empiricals in terms of asymptotic MSE (AMSE), and provide improved confidence bands and hypothesis testing procedures. The properties of the estimators can be proven by simply replacing the DF in EBM [3] by an SDF. However, the confidence band and hypothesis testing procedures in the censored case require different treatments.…”

“…Rojo [18] and Rojo [19] showed that the limiting distributions of the estimators are the same as for the unrestricted case for k = 2 when the linear ordering is strict. El Barmi and Mukerjee [3] (hereafter referred to as EBM [3]) studied the limiting distributions for a general k, and when the ordering is not necessarily strict, providing procedures for asymptotic simultaneous confidence bands and KS type hypothesis tests. It turns out that all of their results hold when the DF's are replaced by SDF's in the uncensored case.…”