  In this letter, the optimality of the likelihood ratio test (LRT) is
investigated for binary hypothesis testing problems in the presence of a
behavioral decision-maker. By utilizing prospect theory, a behavioral
decision-maker is modeled to cognitively distort probabilities and costs based
on some weight and value functions, respectively. It is proved that the LRT may
or may not be an optimal decision rule for prospect theory based binary
hypothesis testing and conditions are derived to specify different scenarios.
In addition, it is shown that when the LRT is an optimal decision rule, it
corresponds to a randomized decision rule in some cases; i.e., nonrandomized
LRTs may not be optimal. This is unlike Bayesian binary hypothesis testing in
which the optimal decision rule can always be expressed in the form of a
nonrandomized LRT. Finally, it is proved that the optimal decision rule for
prospect theory based binary hypothesis testing can always be represented by a
decision rule that randomizes at most two LRTs. Two examples are presented to
corroborate the theoretical results.
