Novel Optimization Methods For Statistical Inference Under Umbrella Restrictions

Statistical inference problem is one of the important subject in statistics.There are usually priori conditions for the amount to be estimated and the hypothesis to be tested.These conditions are used as constraints in statistical inference models to form the sta-tistical inference problem with constraints.The statistical inference with constraints has a wide range of applications in the field of medicine,economic,environmental science and genetics,so it is still a hot research in the statistics world.Dose-response model is a classical order restrictions statistical inference problem.It is a common statistical model to set up the optimal doses which is effective,and has the advantage of less side effects.It is very important for drugs manufacture to handle this problem.In clinical trials,the response-to-dose ratio of drugs does not always increase with the increase of dose,when the dose reaches a certain level,the response-to-dose ratio may be decreased,it obeys the umbrella restrictions.In essence,dose-response model under umbrella restrictions is a constrained optimization problem.Different from statistical method,this article theoretically analyze the dose-response model under umbrella restrictions by optimization theory.Firstly,this paper studied the nature of the dose-response model under umbrella restrictions,proved the existence and uniqueness of its solution,provided the first-order optimality solution and second-order optimality solution,and studied the duality theory of the model.Secondly,this paper provided three algorithms to solve this problem,and also provided the method to determine the umbrella-vertex.At the last,numerical experiment is presented by using real data and a comparison of the three algorithms are given.