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Neyman-pearson criterion

P D , such that P F

The maximization is over all decision rules (equivalently, over all decision regions R 0 , R 1 ). Using different terminology, the Neyman-Pearson criterionselects the most powerful test of size (not exceeding) .

Fortunately, the above optimization problem has an explicit solution. This is given by the celebrated Neyman-Pearson lemma , which we now state. To ease the exposition, our initial statement of this result onlyapplies to continuous random variables, and places a technical condition on the densities. A more general statement is givenlater in the module.

Neyman-pearson lemma: initial statement

Consider the test 0 : x f 0 x 1 : x f 1 x where f i x is a density. Define x f 1 x f 0 x , and assume that x satisfies the condition that for each , x takes on the value with probability zero under hypothesis 0 . The solution to the optimization problem in is given by x f 1 x f 0 x 0 1 where is such that P F x x x f 0 x If 0 , then . The optimal test is unique up to a set of probability zero under 0 and 1 .

The optimal decision rule is called the likelihood ratio test . x is the likelihood ratio , and is a threshold . Observe that neither the likelihood ratio nor the threshold depends on the a priori probabilities i . they depend only on the conditional densities f i and the size constraint . The threshold can often be solved for as a function of , as the next example shows.

Continuing with , suppose we wish to design a Neyman-Pearson decision rule withsize constraint . We have

x 1 2 x 1 2 2 1 2 x 2 2 x 1 2
By taking the natural logarithm of both sides of the LRT and rarranging terms, the decision rule is not changed, and weobtain x 0 1 1 2 Thus, the optimal rule is in fact a thresholding rule like we considered in . The false-alarm probability was seen to be P F Q Thus, we may express the value of required by the Neyman-Pearson lemma in terms of : Q

Sufficient statistics and monotonic transformations

For hypothesis testing involving multiple or vector-valued data, direct evaluation of the size( P F ) and power ( P D ) of a Neyman-Pearson decision rule would require integrationover multi-dimensional, and potentially complicated decision regions. In many cases, however, this can be avoided bysimplifying the LRT to a test of the form t 0 1 where the test statistic t T x is a sufficient statistic for the data. Such a simplified form is arrived at by modifying both sides of the LRT withmontonically increasing transformations, and by algebraic simplifications. Since the modifications do not change thedecision rule, we may calculate P F and P D in terms of the sufficient statistic. For example, the false-alarm probability may be written

P F declare 1 t t t f 0 t
where f 0 t denotes the density of t under 0 . Since t is typically of lower dimension than x , evaluation of P F and P D can be greatly simplified. The key is being able to reduce theLRT to a threshold test involving a sufficient statistic for which we know the distribution .

Common variances, uncommon means

Let's design a Neyman-Pearson decision rule of size for the problem 0 : x 0 2 I 1 : x 1 2 I where 0 , 2 0 are known, 0 0 0 , 1 1 1 are N -dimensional vectors, and I is the N N identity matrix. The likelihood ratio is

x n 1 N 1 2 2 x n 2 2 2 n 1 N 1 2 2 x n 2 2 2 n 1 N x n 2 2 2 n 1 N x n 2 2 2 1 2 2 n 1 N 2 x n 2 1 2 N 2 2 n 1 N x n
To simplify the test further we may apply the natural logarithm and rearrange terms to obtain t n 1 N x n 0 1 2 N 2
We have used the assumption 0 . If 0 , then division by is not a monotonically increasing operation, and the inequalitieswould be reversed.
The test statistic t is sufficient for the unknown mean. To set the threshold , we write the false-alarm probability (size) as P F t t t t f 0 t To evaluate P F , we need to know the density of t under 0 . Fortunately, t is the sum of normal variates, so it is again normally distributed. In particular, we have t A x , where A 1 , so t A 0 A 2 I A 0 N 2 under 0 . Therefore, we may write P F in terms of the Q-function as P F Q N The threshold is thus determined by N Q Under 1 , we have t A 1 A 2 I A N N 2 and so the detection probability (power) is P D t Q N N Writing P D as a function of P F , the ROC curve is given by P D Q Q P F N The quantity N is called the signal-to-noise ratio . As its name suggests, a larger SNR corresponds to improved performance of the Neyman-Pearsondecision rule.
In the context of signal processing, the foregoing problem may be viewed as the problem of detecting aconstant (DC) signal in additive white Gaussian noise : 0 : x n w n , n = 1 , , N 1 : x n A w n , n = 1 , , N where A is a known, fixed amplitude, and w n 0 2 . Here A corresponds to the mean in the example.

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research.net
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Source:  OpenStax, Signal and information processing for sonar. OpenStax CNX. Dec 04, 2007 Download for free at http://cnx.org/content/col10422/1.5
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