4.6 Sequential hypothesis testing  (Page 3/3)

Usually, the likelihood is manipulated to derive a sufficient statistic. The resulting sequential decision rule is $$\begin{array}{cc}{}_L(r)< {}_0(L)& \text{say}{}_0\\ {}_0(L)< {}_L(r)< {}_1(L)& \text{say "need more data"}\\ {}_1(L)< {}_L(r)& \text{say}{}_1\end{array}$$ Note that the thresholds ${}_0(L)$ and ${}_1(L)$ , derived from the thresholds ${}_0$ and ${}_1$ , usually depend on the number of observations used in the decision rule.

Average number of required observations

The awake reader might wonder whether that the sequential likelihood ratio test just derived has the disturbing propertythat it may never terminate: can the likelihood ratio wander between the two thresholds forever? Fortunately, thesequential likelihood ratio test has been shown to terminate with probability one ( Wald ). Confident of eventual termination, we need to explore how manyobservations are required to meet performance specifications. The number of observations is variable, depending on theobserved data and the stringency of the specifications. The average number of observations required can be determined in the interesting case when theobservations are statistically independent.

Assuming that the observations are statistically independent and identically distributed, the likelihood ratio is equal tothe product of the likelihood ratios evaluated at each observation. Considering $\ln {}_{L_0}(r)$ , the logarithm of the likelihood ratio when a decision is made on observation $L_0$ , we have $$\ln {}_{L_0}(r)=\sum_{l=0}^{L_0-1} \ln (r_l)$$ where $(r_l)$ is the likelihood ratio corresponding to the $l^{\mathrm{th}}$ observation. We seek an expression for $(L_0)$ , the expected value of the number of observations required to make the decision. To derive this quantity, weevaluate the expected value of the likelihood ratio when the decision is made. This value will usually vary with whichmodel is actually valid; we must consider both models separately. Using the laws of conditional expectation (see Joint Distributions ), we find that the expected value of ${}_{L_0}(r)$ , assuming that model ${}_1$ was true, is given by $$({}_1, \ln {}_{L_0}(r))=(({}_1, L_0, \ln {}_{L_0}(r)))$$ The outer expected value is evaluated with respect to the probability distribution of $L_0$ ; the inner expected value is average value of the log-likelihood assuming that $L_0$ observations were required to choose model ${}_1$ . In the latter case, the log-likelihood is the sum of $L_0$ component log-likelihood ratios $$({}_1, L_0, \ln {}_{L_0}(r))=L_0({}_1, \ln (r_l))$$ Noting that the expected value on the right is a constant with respect to the outer expected value, we find that $$({}_1, \ln {}_{L_0}(r))=({}_1, L_0)({}_1, \ln (r_l))$$ The average number of observations required to make a decision, correct or incorrect, assuming that ${}_1$ is true is thus expressed by $$({}_1, L_0)=\frac{({}_1, \ln {}_{L_0}(r))}{({}_1, \ln (r_l))}$$ Assuming that the other model was true, we have the complementary result $$({}_0, L_0)=\frac{({}_0, \ln {}_{L_0}(r))}{({}_0, \ln (r_l))}$$

The numerator is difficult to calculate exactly but easily approximated; assuming that the likelihood ratio equals itsthreshold value when the decision is made, $$({}_0, \ln {}_{L_0}(r))\approx P_F\ln {}_1-1\ln {}_0{P}_F\ln \left(\frac{P_D}{P_F}\right)-1\ln \left(\frac{1-P_D}{1-P_F}\right)$$ $$({}_1, \ln {}_{L_0}(r))\approx P_D\ln {}_1-1\ln {}_0{P}_D\ln \left(\frac{P_D}{P_F}\right)-1\ln \left(\frac{1-P_D}{1-P_F}\right)$$ Note these expressions are not problem dependent; they depend only on the specified probabilities.The denominator cannot be approximated in a similar way with such generality; it must be evaluated for each problem.

We must not forget that these results apply to the average number of observations required to make a decision. Expressions for the distribution of thenumber of observations are complicated and depend heavily on the problem. When an extremely large number of observationare required to resolve a difficult case to the required accuracy, we are forced to truncate the sequential test, stopping when a specified number ofobservations have been used. A decision would then be made by dividing the region between the boundaries in half andselecting the model corresponding to the boundary nearest to the sufficient statistic. If this truncation point is largerthan the expected number, the performance probabilities will change little. "Larger" is again problem dependent;analytic results are few, leaving the option of computer simulations to estimate the distribution of the number ofobservations required for a decision.