In the present paper, for constructing the minimum risk estimators of state of stochastic systems, a new technique of invariant embedding of sample statistics in a loss function is proposed. This technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors. It allows one to eliminate unknown parameters from the problem and to find the best invariant estimator, which has smaller risk than any of the well‐known estimators. There exists a class of control systems where observations are not available at every time due to either physical impossibility and/or the costs involved in taking a measurement. In this paper, the problem of how to select the total number of the observations optimally when a constant cost is incurred for each observation taken is discussed. To illustrate the proposed technique, an example is given and comparison between the maximum likelihood estimator (MLE), minimum variance unbiased estimator (MVUE), minimum mean square error estimator (MMSEE), median unbiased estimator (MUE), and the best invariant estimator (BIE) is discussed.
One of the most important problems in fatigue analysis and design of aircraft structures is the prediction of fatigue crack growth in service. Available in‐service inspection data for various types of aircraft indicate that the fatigue crack damage accumulation in service involves considerable statistical variability. In this paper, we consider the problem of estimating the minimum time to crack initiation (or warranty period) for a number of aircraft structural components, before which no cracks (that may be detected) in materials occur, based on the results of previous warranty period tests on the structural components in question. This problem is a special case of a general class of problems concerned with the analysis of fatigue crack damage accumulation in aircraft service. The technique proposed here for solving this problem emphasizes pivotal quantities relevant for obtaining ancillary statistics. Attention is restricted to invariant families of distributions. Numerical examples are given.
In this paper, the following biomedical problem is considered. People are subjected to a certain chemotherapeutic treatment. The optimal dosage is the maximal dose for which an individual patient will have toxicity level that does not cross the allowable limit. We discuss sequential procedures for searching the optimal dosage, which are based on the concept of dual control and the principle of optimality. According to the dual control theory, the control has two purposes that might be conflicting: one is to help learning about unknown parameters and/or the state of the system (estimation); the other is to achieve the control objective. Thus the resulting control sequence exhibits the closed‐loop property, i.e. it anticipates how future learning will be accomplished and how it can be fully utilized. Thus, in addition to being adaptive, this control also plans its future learning according to the control objective. Results are obtained for a priori uniform distribution of the unknown dosage. Because answers can be obtained analytically without approximation, the optimum policy can be compared with the non‐optimum policy of optimizing stage by stage. An illustrative example is given.
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