2.10 Non-random parameters

Continuing the example of the previous section , let's consider the situation where the value of the mean of each observationunder model ${}_1$ is not known. The sufficient statistic is the sum of the observations (that quantity doesn't depend on $m$ ) and the distribution of the observation vector under model ${}_0$ does not depend on $m$ (allowing computation of the false-alarm probability). However, a subtlety emerges; in the derivation of thesufficient statistic, we had to divide by the value of the mean. The critical step occurs once the logarithm of thelikelihood ratio is manipulated to obtain $$m\sum_{l=0}^{L-1} r_l\underset{{}_0}{\overset{{}_1}{}}(^2\ln +\frac{Lm^2}{2})$$ Recall that only positively monotonic transformations can be applied; if a negatively monotonicoperation is applied to this inequality (such as multiplying both sides by -1), the inequality reverses . If the sign of $m$ is known, it can be taken into account explicitly and a sufficient statistic results. If, however, the sign is notknown, the above expression cannot be manipulated further and the left side constitutes the sufficient statistic for thisproblem. The sufficient statistic then depends on the unknown parameter and we cannot develop a decision rule in this case.If the sign is known, we can proceed. Assuming the sign of $m$ is positive, the sufficient statistic is the sum of the observations and the threshold $$ is found by $$=\sqrt{L}Q^{(-1)}(P_F)$$ Note that if the variance $^2$ instead of the mean were unknown, we could not compute the threshold. The difficulty lies not with the sufficientstatistic (it doesn't depend on the variance), but with the false alarm probability as the expression indicates. Anotherapproach is required to deal with the unknown-variance problem.