Hi,
I'm starting to read about statistical power analysis for a problem
related to an engineering application. I went back and reviewed a
****tion of a *classic* textbook, "Statistical Signal Processing" by
Loius Scharf and came across a statement that floored me.
Scharf is talking about the decision rules for a simple binary
hypothesis test:
H0 : Theta = Theta_0
H1 : Theta = Theta_1
He then discusses a curve he calls the "Receiver Operating
Characteristics", which is a curve of alpha vs beta (or as he puts it,
the Prob of a False Alarm vs the Prob of Detection). The curve Scharf
plots is a continuous convex curve that goes from (0,1) to (1,1).
Here is the statement Scharf makes next," If the prob of false alarm
equals zero, then H0 is always selected, meaning that H1 is never
selected, and that the probability of detection equals zero." (or, the
way I would read it is, alpha=0 implies beta = 0).
I would strongly disagree with his statement that "If the prob of
false alarm equals zero, then H0 is always selected". The prob of a
false alarm is a conditional prob, i.e. it is the prob I choose H1
given H0 is true. Thus to say that the prob of a false alarm equals
zero, means, to me, that H0 is always chosen, given H0 is true. Or to
put it another way, in general, alpha = 0 tells me nothing about what
decision I will make if the conditional hypothesis is not satisfied,
i.e. if H1 is true.
OT1H, the text is a classic and the ROC curves are well-known in
engineering, and it's hard to believe he made a mistake, but OTOH, I
read this, and it just seems plain wrong.
Any suggestions about who is right or if I am misreading something or
misunderstanding Scharf?
Thank-you,
Matt
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