Universal composite hypothesis testing: a competitive minimax approach

A novel approach is presented for the long-standing problem of composite hypothesis testing. In composite hypothesis testing, unlike in simple hypothesis testing, the probability function of the observed data, given the hypothesis, is uncertain as it depends on the unknown value of some parameter. The proposed approach is to minimize the worst case ratio between the probability of error of a decision rule that is independent of the unknown parameters and the minimum probability of error attainable given the parameters. The principal solution to this minimax problem is presented and the resulting decision rule is discussed. Since the exact solution is, in general, hard to find, and a fortiori hard to implement, an approximation method that yields an asymptotically minimax decision rule is proposed. Finally, a variety of potential application areas are provided in signal processing and communications with special emphasis on universal decoding.