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Extra resources for Advances in Computers, Vol. 21
Since H4 provides little information concerning f(xil H,), the density f ( x i l H 4 ) should provide as little help as possible for testing against f(xilK4);f ( x i l H 4 ) should therefore be as close tof(xilK4) as possible. 3-5) f(XiIH4) = 2 (f' + f - ) The density f ( x i l H 4 ) has zero median, and hence belongs to H,; and f(x,lH,) closely resembles f ( x i l K 4 ) since f ( x i l H 4 ) has the same shape as If(x,IK4) on the positive and negative axes. 3-6) The ratio of the signal present density to the signal absent density in Eq.
3-5) f(XiIH4) = 2 (f' + f - ) The density f ( x i l H 4 ) has zero median, and hence belongs to H,; and f(x,lH,) closely resembles f ( x i l K 4 ) since f ( x i l H 4 ) has the same shape as If(x,IK4) on the positive and negative axes. 3-6) The ratio of the signal present density to the signal absent density in Eq. 3-8) If n, = Z~,,u(x,)is used to denote the number of positive observations, the likelihood ratio may be written using Eqs. 3-9) L ( x ) = (2p)"' [2(1 - p)]"-"+ Eq. 3-12) H4 Since it can be shown that the detector in Eq.
Unbiasedness is a simple and reasonable condition to impose on a detector. 6 Composite Hypotheses 29 F E K is never less than the level of the test. The reasonableness of requiring a test to be unbiased can be seen by noting that if the conditions given in Eq. 6-10) are not satisfied, the probability of accepting H will be greater under some distributions contained in the alternative K than in some cases when the hypothesis H is true. 6-11) Although Eq. 6-10) states the concept of unbiasedness quite clearly, it is difficult to employ this definition to find most powerful unbiased tests because of the presence of the inequalities.
Advances in Computers, Vol. 21