Page 272 - Schaum's Outlines - Probability, Random Variables And Random Processes

CHAP. 81 DECISION THEORY 265

where Di (i = 0, 1) denotes the event that the decision is made to accept Hi. PI is often denoted by a
and is known as the level of signijcance, and PI, is denoted by fl and (1 - /3) is known as the power of
the test. Note that since a and /? represent probabilities of events from the same decision problem,
they are not independent of each other or of the sample size n. It would be desirable to have a
decision process such that both a and fl will be small. However, in general, a decrease in one type of
error leads to an increase in the other type for a fixed sample size (Prob. 8.4). The only way to
simultaneously reduce both type of errors is to increase the sample size (Prob. 8.5). One might also
attach some relative importance (or cost) to the four possible courses of action and minimize the total
cost of the decision (see Sec. 8.3D).
The probabilities of correct decisions (actions 1 and 3) may he expressed as

In radar signal detection, the two hypotheses are
H, : No target exists
HI : Target is present
In this case, the probability of a Type I error PI = P(Dl I H,) is often referred to as the false-alarm
probability (denoted by P,), the probability of a Type I1 error PI, = P(Do I HI) as the miss probability
(denoted by P,), and P(Dl (HI) as the detection probability (denoted by PD). The cost of failing to
detect a target cannot be easily determined. In general we set a value of P, which is acceptable and
seek a decision test that constrains P, to this value while maximizing P, (or equivalently minimizing
P,). This test is known as the Neyman-Pearson test (see Sec. 8.3C).

8.3 DECISION TESTS
A. Maximum-Likelihood Test :
Let x be the observation vector and P(x I Hi), i = 0.1, denote the probability of observing x given
that Hi was true. In the maximum-likelihood test, the decision regions R, and R, are selected as
Ro = {x: P(x I H,) > P(x ( HI)}
Rl = (x: P(x I Ho) < P(x I HI))
Thus, the maximum-likelihood test can be expressed as

if P(x ( H,) > P(x I HI)
d(x)={i: ifP(xHo)<P(xHl)
The above decision test can be rewritten as

If we define the likelihood ratio A(x) as

then the maximum-likelihood test (8.7) can be expressed as