On the design of extended Neyman-Pearson hypothesis tests

We consider a problem in which exactly one of n+1 distinct signals {S₀, ..., S_n}, say S₀, may be present in a noisy environment and the presence or absence of S₀ needs to be determined. The performance is given by the false alarm probabilities P_f(i), i = 1, ..., n, where P_f(i) is the probability that S_i is declared as S₀, and the detection probability P_d which is the probability that S₀ is correctly recognized. It is required that P_d be maximized while P_f(i)≤c_i, i = 1, ..., n, where c₁, ..., c_n are prescribed non-negative constants. The solution is an extended Neyman-Pearson test in which p₀(x) is tested against (w₁p₁(x)+...+w_np_n(x)) where p_i(x) is the probability density function of observation X when S_i occurs. We propose two methods to obtain w₁, ..., w_n for any given c₁, ..., c_n. For certain simple cases, analytical or graphical solution is used. For the general case, we propose a search algorithm on w₁, ..., w_n which provides a sequence of extended Neyman-Pearson tests that converge to the optimal solution. Numerical examples are given to illustrate these methods.