经典复合假设检验 【授课PPT】【第一部分：介绍】.pdfVIP

(An Introduction to) Classical Composite Hypothesis Testing First, recall that, in the composite testing case, we have Θ0 and Θ1 that form a partition of the parameter space Θ: Θ0 ∪ Θ1 = Θ, Θ0 ∩ Θ1 = ∅ and we wish to identify which of the following two hypotheses is true: H : θ ∈ Θ , null hypothesis 0 0 H : θ ∈ Θ , alternative hypothesis. 1 1 Here, we adopt the classical Neyman-Pearson approach — maximize the detection probability for a specified false-alarm rate. Example: Suppose that we wish to detect an unknown positive DC level A (A 0): H0 : x [n] = w [n], n = 1, 2, . . . , N H1 : x [n] = A + w [n], n = 1, 2, . . . , N where w [n] is zero-mean white Gaussian noise with known variance σ2 . Here is an alternative formulation: Consider this Detection and Estimation Theory, # 7 1 family of probability density functions (pdfs): N p (x ; θ) = 1 2 N · exp − 12 (x [n] − θ)2 (1) (2πσ ) 2σ n=1 and the following (equivalent) hypotheses: H0 : θ = 0 (signal absent), Θ0 = {0} versus H1 : θ = A 0 (signal present), Θ1 = (0, ∞) where A is unknown, except for its sign. Let us try the classical Neyman-Pearson approach (which required the exact knowledge of A since we considered only simple hypotheses under the classical setting, until now): 2 N/2 1 N 2 1/(2πσ ) · exp[−2σ2 n=1 (x [n] − A) ] Λ(x) =