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_15: The chief aim of this paper is to provide a short technical summary of the principal noise-model results (11-[11], developed by the author to date, and to consider their major implications for 1) measurement, and 2) performance prediction and evaluation. The discussion is illustrated by a concise review of model statistics, methods of estimating the model parameters, including the effects on accuracy of finite data samples. A variety of signal detection problems are similarly used to illustrate the very large potential improvement possible when optimum algorithms are employed, vis-a-vis most of the currently used systems, which are conventionally optimized for Gaussian interference. Conditions for realizing such gains are also given, and the principal key properties of these new, general models are discussed. These models make possible, for the first time, a systematic treatment of real-world EMI environments, both for measurement and assessment, and for receiver performance therein.
@cite_14: 
@cite_7: This paper considers the detection of the presence absence of signals in uncertain low SNR environments. Small modeling uncertainties are unavoidable in any practical system and so robustness to them is a fundamental performance metric. The impact of these modeling uncertainties can be quantified by the position of the "SNR wall" below which a detector will fail to be robust, no matter how long it can observe the channel. We propose simple mathematical models for the uncertainty in the noise and fading processes. These are used to show what aspects of the model lead to SNR walls for differing levels of knowledge of the signal to be detected. These results have implications for wireless spectrum regulators. The context is opportunistically sharing spectrum with primary users that must be detected in order to avoid causing harmful interference on a channel. Ideally, a secondary system would be able to detect primaries robustly without having to know much about their signaling strategies. We argue that the tension between primary and secondary users is captured by the technical question of computing the optimal tradeoff between the primary user's capacity and the secondary user's sensing robustness as quantified by the SNR wall. This is an open problem, but we compute this tradeoff for some simple detectors.
Similar problems are also studied in the application of the signal detection in noise with uncertainty @cite_7 or non-Gaussian noise @cite_15 @cite_14 . Noise uncertainty considers the case where the test designers do not know the noise statistics perfectly. For example, it might be known that the mean or the variance of the noise falls onto an interval but its exact value may be unknown. Non-Gaussian noise considers the case where the noise statistics is a mixture of Gaussian densities of various means and variances. The distribution function constraint can be applied whenever robustness is needed. Besides applications in signal processing, the proposed test has many practical applications, such as quality assurance in manufacturing, event forecasting, assessment of model fitting in finance, to name just a few.