4.16 Robust hypothesis testing  (Page 2/2)

Letting $p^{}$ denote the worst-case density, out minimax procedure results in the following densities for each model in thelikelihood ratio test. $$p^{}(r_l, {}_i, r_l)=\begin{cases}{p}^o(r_l^{}, {}_0, r_l^{})C_i^{}e^{-(K^{}\left|r_l-r_l^{}\right|)} & \text{if $r_l< r_l^{}$}\\ p^o(r_l, {}_i, r_l) & \text{if $r_l^{}< r_l< r_l^{}$}\\ p^o(r_l^{}, {}_0, r_l^{})C_i^{}e^{-(K^{}\left|r_l-r_l^{}\right|)} & \text{if $r_l> r_l^{}$}\end{cases}$$ The constants $K^{}$ and $K^{}$ determine the rate of decay of the exponential tails of these worst-case distributions. Their specific values havenot yet been determined, but since they are not needed to compute the likelihood ratio, we don't need them. The constants $C_i^{}$ and $C_i^{}$ are required so that a unit-area density results. The likelihood ratio for each observation in the robust model evaluationproblem becomes

The evaluation of the likelihood ratio depends entirely on determining values for $r_l^{}$ and $r_l^{}$ . The ratios $\frac{C_1^{}}{C_0^{}}=c^{}$ and $\frac{C_1^{}}{C_0^{}}=c^{}$ are easily found; in the tails, the value of thelikelihood ration equals that at the edges of the central region for continuous densities. $$c^{}=\frac{p^o(r_l, {}_1, r_l^{})}{p^o(r_l, {}_0, r_l^{})}$$ $$c^{}=\frac{p^o(r_l, {}_1, r_l^{})}{p^o(r_l, {}_0, r_l^{})}$$ At the left boundary, for example, the distribution functions must satisfy $(1-)p(r_l, {}_0, r_l^{})=1p(r_l, {}_1, r_l^{})+$ . In terms of the nominal densities, we have $$\int_{()} \,d x$$∞ r l p r l 0 x p r l 1 x 1 This equation also applies the value right edge $r_l^{}$ . Thus, for a given value of $$ , the integral of the difference between the nominal densities should equal the ratio $\frac{}{1-}$ for two values. illustrates this effect for a Gaussian example. The bi-valued nature of thisintegral may not be valid for some values of $$ ; the value chosen for $$ can be too large, making it impossible to distinguish the models! This unfortunatecircumstance means that the uncertainties, as described by the value of $$ , swamp the characteristics that distinguish the models. Thus, the modelsmust be made more precise (more must be known about the data) so that smaller deviations from the nominal models can describe theobservations.

Returning to the likelihood ratio, the "robust" decision rule consists of computing a clipped function of each observed value, multiplying them together, and comparingthe product computed over the observations with a threshold value. We assume that the nominal distributions of each of the $L$ observations are equal; the values of the boundaries $r_l^{}$ and $r_l^{}$ then do not depend on the observation index $l$ in this case. More simply, evaluating the logarithm of the quantities involved results inthe decision rule $$\sum_{l=0}^{L-1} f(r_l)\underset{{}_0}{\overset{{}_1}{}}$$ where the function $f()$ is the clipping function given by $$f(r_l)=\begin{cases}\ln c^{} & \text{if $r_l< r^{}$}\\ \ln \left(\frac{p^o(r_l, {}_1, r_l)}{p^o(r_l, {}_0, r_l)}\right) & \text{if $r^{}< r_l< r^{}$}\\ \ln c^{} & \text{if $r^{}< r_l$}\end{cases}$$ If the observations were not identically distributed, then the clipping function would depend on the observation index.

Determining the threshold $$ that meets a specific performance criterion is difficult in the context of robust model evaluation. By the very nature of theproblem formulation, some degree of uncertainty in the a priori densities exists. A specific false-alarm probability can be guaranteed by using theworst-case distribution under ${}_0$ . This density has the disturbance term begin an impulse at infinity. Thus, the expected value $m_c$ of a clipped observation $f(r_l)$ with respect to the worst-case density is $1(f(r_l))+\ln c^{}$ where the expected value in this expression is evaluated with respect to the nominal density under ${}_0$ . Similarly, an expression for the variance ${}_c^2$ of the clipped observation can be derived. As the decision rule computes the sum of the clipped, statisticallyindependent observations, the Central Limit Theorem can be applied to the sum, with the result that the worst-casefalse-alarm probability will approximately equal $Q(\frac{-L{m}_c}{\sqrt{L}{}_c})$ . The threshold $$ can then be found which will guarantee a specified performance level. Usually, the worst-case situation does not occur and thethreshold set by this method is conservative. We can assess the degree of conservatism by evaluating these quantities under thenominal density rather than the worst-case density.

Let's consider the Gaussian model evaluation problem we have been using so extensively. The individual observations arestatistically independent and identically distributed with variance five: $^2=5$ . For model ${}_0$ , the mean is zero; for ${}_1$ , the mean is one. These nominal densities describe our best models for the observations, but we seek to allowslight deviations (10%) from them. The equation to be solved for the boundaries is the implicit equation $$Q(\frac{z-m}{})-Q(\frac{z}{})=\frac{}{1-}$$ The quantity on the left side of the equation is shown in . If the uncertainty in the Gaussian model, as expressed by the parameter $$ , is larger than 0.15 (for the example values of $m$ and $$ ), no solution exists. Assuming that $$ equals 0.1, the quantity $\frac{}{1-}=0.11$ and the clipping thresholds are $r^{}=-1.675$ and $r^{}=2.675$ . Between these values, the clipping function is given by the logarithm of the likelihood ratio, which is givenby $\frac{2m{r}_l-m^2}{2^2}$ .

We can decompose the clipping operation into a cascade of two operations: a linear scaling and shifting (as described by theprevious expression) followed by a clipper having unit slope (see ).