This paper presents new mathematical models and methods for signal detection in correlated non-Gaussian noise with excess kurtosis, which commonly arises in radar, communication, and monitoring systems. Traditional Gaussian-based detection methods and full probability density function (PDF) approaches become impractical or inaccurate in such environments. To address this, the paper proposes the use of higher-order statistics (HOS), specifically one-dimensional and two-dimensional moment-cumulant models, which effectively describe both non-Gaussianity and statistical dependencies. Polynomial stochastic decision rules (DRs) are synthesized and optimized using a modified moment-based quality criterion that estimates upper bounds for Type I and Type II error probabilities. Linear and nonlinear DRs are analyzed; the latter include higher-order moments such as excess kurtosis to better capture noise structure. Simulation results show that nonlinear DRs provide improved detection performance, especially in heavy-tailed distributions and strong correlations. The proposed approach enhances detection accuracy and reduces error rates without requiring full PDF modeling. These results demonstrate the effectiveness of HOS-based methods for intelligent signal processing and support their practical use in complex noise environments.