Flexible Distributional Modeling with the ACT-G Family Using Recurrent Neural Networks and Firefly Optimization

Skewed and heavy-tailed data modeling has continued to be the focal issue in survival, reliability, and applied sciences, because current distribution families are often rigid or difficult to compute. We address this gap with a new family, the arc-cosine-tangent-generators family, which extends sine-based and exponentiated-based generators by a relatively simple transformation. The most significant contribution of the research is the presentation of the ACT-G family, providing more flexibility of tail and skewness control without new shape parameters, and with closed-form expressions of the density, distribution, and hazard functions. As a special case, we derive the ACTE Extreme Value distribution and obtain its statistical properties. One is maximum likelihood estimation and a hybrid machine learning solution relying on recurrent neural networks and Firefly Algorithm to overcome convergence issues on high-dimensional likelihood surfaces. Experiments in simulation were performed with different sample sizes and different parameters, and then applied to music therapy data. Findings indicate that ACTE has always performed better than the ASTE-exponential and traditional maximum likelihood estimators, with lower bias and mean squared error, better tail fit and better robustness. It is also adequately supported by information criteria (AIC, BIC) and goodness-of-fit tests (KS, AD). Finally, the ACT-G family is a statistically rigorous and practically useful model of skewed and heavy tailed data, and, naturally, which also has some advantages in simulation and in the real world.