In this work, we study non-parametric hypothesis testing problem with distribution function constraints. The empirical likelihood ratio test has been widely used in testing problems with moment (in)equality constraints. However, some detection problems cannot be described using moment (in)equalities. We propose a distribution function constraint along with an empirical likelihood ratio test. This detector is applicable to a wide variety of robust parametric non-parametric detection problems. Since the distribution function constraints provide a more exact description of the null hypothesis, the test outperforms the empirical likelihood ratio test with moment constraints as well as many popular goodness-of-fit tests, such as the robust Kolmogorov-Smirnov test and the Cramer-von Mises test. Examples from communication systems with real-world noise samples are provided to show their performance. Specifically, the proposed test significantly outperforms the robust Kolmogorov-Smirnov test and the Cramer-von Mises test when the null hypothesis is nested in the alternative hypothesis. The same example is repeated when we assume no noise uncertainty. By doing so, we are able to claim that in our case, it is necessary to include uncertainty in noise distribution. Additionally, the asymptotic optimality of the proposed test is provided.
@cite_18: The generalized likelihood ratio test (GLRT), which is commonly used in composite hypothesis testing problems, is investigated. Conditions for asymptotic optimality of the GLRT in the Neyman-Pearson sense are studied and discussed. First, a general necessary and sufficient condition is established, and then based on this, a sufficient condition, which is easier to verify, is derived. A counterexample where the GLRT is not optimal, is provided as well. A conjecture is stated concerning the optimality of the GLRT for the class of finite-state sources. >
@cite_22: This paper considers the design of a minimax test for two hypotheses where the actual probability densities of the observations are located in neighborhoods obtained by placing a bound on the relative entropy between actual and nominal densities. The minimax problem admits a saddle point which is characterized. The robust test applies a nonlinear transformation which flattens the nominal likelihood ratio in the vicinity of one. Results are illustrated by considering the transmission of binary data in the presence of additive noise.
@cite_2: 
The problem under study in the paper covers a wide range of applications. Several studies in the literature are closely related to this work. As previously discussed, Huber's robust detection problem can be treated as a special case of our problem. In @cite_2 , Huber's test features a clipped version of the likelihood ratio test between the nominal densities that delivers performance which minimizes the worst-case probability of false alarm and miss. However, the clipped test has limited application since it requires parametric model for the nominal probability distributions. In @cite_22 , the author provides a framework of the robust likelihood ratio test when the uncertainty region is described by the Kullback-Leibler divergence. In the parametric case, this problem is also known as the composite hypothesis testing problem [p. 169] levy2008principles . In this case, the generalized likelihood ratio test (GLRT) has optimal error performance when the probability distributions under all the hypotheses are in the same exponential family, c.f. [p. 204] levy2008principles , @cite_18 . Again, this test requires complete parametric description of the probability distributions.