Likelihood-based Approaches for Inequality Restrictions Problems with or without Incomplete Data

Structure involving inequalities is often used in the statistical inference. Parameter estimation involving inequality is very common in many applications. For example, the response probability of a particular treatment may increase with the dosage level; or the treatment response may stochastically dominate the control. In fact, utilization of such inequality ordering information increases the eciency of statistical inference procedures (see, Barlow et al., 1972; and Robertson et al., 1988; Silvapulle and Sen, 2004). Especially, Dykstra et al. (2002) edited a special issue for the Journal of Statistical Planning and Inference to discuss statistical inference under inequality restrictions. It is well known that, in this setting, the parameter space is not open. Consequently, it is dicult to nd the maximum likelihood estimates when the true parameters lie on the boundary of the parameter space. Shi (1994) and Shi and Jiang (1998) considered the problem with order restrictions on normal distribution parameters. Existing statistical research for inequality restrictions is mainly for complete data. However, in clinical trials and observational studies, complete data are often not available from every subject due to the loss of records, the unavailability of covariate measurements, or survey nonresponse. Amongst, the EM algorithm, proposed by Dempster, Laird and Rubin (1977) is one of the most popular procedures for obtaining maximum likelihood estimates for many incomplete-data problems. When the parameters must satisfy a set of restrictions, application of the EM algorithm may not be directly applicable. Kim and Taylor (1995) propose