In this work, we focus on mixtures with negative coefficients and their applications in computational statistics. Mixtures of probability densities are widely used in statistics and machine learning. While classical mixtures restrict weights to be non-negative, allowing negative weights enables more flexible density approximation. However, negative weights introduce challenges in handling and sampling such distributions. For this purpose, we propose efficient Monte Carlo (MC) methods (including MC quadratures, rejection sampling and importance sampling schemes) for computing integrals and generating samples from these mixtures. A tailored proposal density ensures accurate and efficient generation of (unweighted) samples. Furthermore,we introduce an IS scheme which employs a mixture with negative coefficients as a proposal density, yielding samples with both positive and negative importance weights. Applications in Gaussian process-based density estimation demonstrate the practical relevance and efficiency of proposed schemes. An adaptive importance sampling procedure based on GP-regression is also proposed. The numerical results provide clear empirical evidence of the accuracy and computational efficiency of the proposed methods.