4.4
Distributions and Uncertainty Sets Depending
on the Test Statistic
In the design of minimax robust sequential tests, distributions arise that are parame- terized in the test statistic, meaning that the distribution of Xn depends on the value
of the test statistic Tn−1. The notation
PXn+1|x1,...,xn = PXn+1|tn (4.6) is used to indicate this dependence. For test statistics of the form (3.15), which will be shown to be optimal in the minimax case as well, this implies that PXn not only depends on θ, but also on the likelihood ratios z0 and z1, i.e.,
PXn+1|x1,...,xn = PXn+1|zn,θn. (4.7) Irrespective of the additional complexity, optimal sequential tests for distributions de- pending on the test statistic can still be analyzed and designed within the framework presented in the previous chapter, namely, by choosing the sufficient statistic for the distribution of Xn+1 as
˜
θn:= (zn, θn). (4.8)
Consequently, the results obtained in the previous sections can be extended to distri- butions depending on the test statistic by simply substituting ˜θ for θ in the respective equations. Interestingly, for sufficient statistics of the form (4.8), no additional test statistic is required, or, more precisely, the test statistic and the Markov statistic co- incide so that Tn = ˜Θ
n. Nevertheless, the explicit distinction between z and θ is
maintained in the upcoming sections, meaning that the conditional distributions are written as Pz,θ.
A result that is of particular importance for the derivation of minimax robust sequential tests is the function ρλ in Theorem 5. Allowing P0, P1and P to depend on z in addition
to θ, (3.12) becomes ρλ(z, θ) = min gλ(z) , 1 + Z ρλ z0 p0z,θ(x) pz,θ(x) , z1 p1z,θ(x) pθ(x) , ξz,θ(x) dPz,θ(x) . (4.9) This definition of ρλ is used repeatedly in the upcoming sections.
assumption. Given the permutation invariance of the fixed sample size test (the outcome of the test does not depend on the ordering of the samples), the i.i.d. property is intuitively reasonable. For the sequential test, however, this symmetry argument no longer holds since rearranging the samples can indeed alter the outcome of the test—compare ordering the observations by significance for the alternative hypotheses to ordering the samples by significance for the null hypothesis.
Another central result that carries over to distributions depending on the test statistic are the integral equations (2.41)–(2.43) in Section 2.4.3. The extension follows imme- diately from the fact that Tn= (Zn, Θn) still satisfies property (2.24), i.e., it provides
all information necessary to determine the conditional distribution of Tn+1.
Given that θ is a component of the test statistic, the uncertainty sets Pθ0, Pθ1 and Pθ
are functions of the test statistic as well. In order to present results for tests based on test statistics different form (z, θ), the generic notations Pt0, Pt1 and Pt are used to
indicate this dependence. In general, this notation should be read in the sense that the test statistic provides sufficient information to determine the uncertainty set for the distribution of the random variable generating the next sample, i.e.,
PXn+1|x1,...,xn−1 = PXn+1|θn = PXn|tn.
Although incorporating dependencies of the form (4.6) in the optimal sequential testing framework is technically straightforward, the question why the behavior of a stochastic process should depend on the test statistic deserves a more detailed answer. Put another way: How can the test design have an influence on the process that is being tested? Even though examples exist for which the testing procedure does indeed affect the stochastic process,2 in typical signal processing applications the distribution of X is entirely independent of the test design. Nevertheless, state dependent distributions are a crucial concept for the design of minimax robust tests, irrespective of whether or not they arise in practice. The reason for this is that the least favorable distribution of the random variable generating the next observation depends on the current state of the test statistic. Assume, for example, that the test statistic is close to the stopping region corresponding to a decision for H0. In this state, another observation in favor of
H0 most likely causes the test to stop. Hence, a distribution that is least favorable with
respect to the expected run-length, needs to place as much probability mass as possible on observations in favor of H1. The opposite is true if the test statistic is close to
the stopping region corresponding to a decision for H1. In this case, the distribution
that maximizes the expected run-length is concentrated on observations in favor of H0.
This idea is illustrated in Figure 4.1 using the example of the sequential probability ratio test with constant thresholds. The red arrows indicate the direction of the drift that is least favorable in the current state.
2A typical example is a test for the efficacy of a new drug or treatment. If such a test is based
on features that the participant has access to (blood pressure, eyesight, allergic reactions, etc.), this knowledge can influence the participant’s response to the treatment. It is well known, for example, that good or bad results in the beginning can change a participant’s confidence in the treatment which in turn affects the expected progress [Kne14]. More details on the interesting topic of sequential medical trials can be found in [Cho11] and [BLS13].
4.4 Distributions and Uncertainty Sets Depending on the Test Statistic 71 n z1 A B 1 2 3 4 5 6 δ = 1 δ = 0 n z1 A B 1 2 3 4 5 6 δ = 1 δ = 0
Figure 4.1: Illustration of state dependent least favorable distributions for the se- quential probability ratio test. For z1 close to the upper threshold, observations in
favor of H0 are least favorable with respect to the expected run-length. For z1 close to
the lower threshold, observations in favor of H1 are least favorable.
The same considerations hold true for the error probabilities. Since the set of states that can be reached with the next update of the test statistic depends on its current state, the least favorable distributions with respect to the error probabilities also depend on the current state. In practice, however, the effect is more pronounced for the expected run-length than for the error probabilities. A more detailed discussion of this phenomenon is deferred to Section 5.8.
In light of these considerations, the property that the test statistic coincides with the Markov statistic is a characteristic of minimax procedures. In essence, Tn = ˜Θ
n im-
plies that the test designer and his virtual adversary, who “designs” the stochastic process X, have access to the same information: The sequential test is designed based on knowledge of the Markov statistic, which determines the expected behavior of the stochastic process. The stochastic process, in turn, is designed based on knowledge of the test statistic, which determines the expected performance of the test. This constel- lation is a necessary consequence of the minimax principle: Distributions that do not adapt to the test statistic cannot be guaranteed to be least favorable and testing pro-
cedures that are based on insufficient information about the stochastic process cannot be guaranteed to be optimal.
In the next Chapter, least favorable distributions are introduced in a more formal and precise manner. In order to do so, however, some basics of convex optimization in Banach spaces need to be introduced.