The Cosine Alpha Power-G Family of Distributions: Theory, Properties, and Applications

This paper proposes a new and versatile family of probability distributions, termed the cosine alpha power-G (CAP-G) family. The construction combines the alpha power transformation with a cosine-based generator, yielding models that can flexibly capture skewness, kurtosis, and a broad spectrum of hazard rate shapes. A single shape-controlling parameter governs the tail behavior and overall form of the distribution, enhancing adaptability while avoiding excessive model complexity. Fundamental properties of the CAP-G family are derived, including the density, distribution, hazard and survival functions, quantile function, moments, and entropy measures. Several well-known distributions arise as special cases, and graphical illustrations highlight the ability of the family to produce both monotonic and nonmonotonic shapes. Detailed theoretical and inferential development is provided for the cosine alpha power exponential (CAP-E) submodel as the primary illustrative member of the family. Parameter estimation is carried out via maximum likelihood, and an expanded Monte Carlo simulation study assesses finite-sample performance through bias, mean squared error, coverage probabilities, and confidence interval widths across multiple parameter configurations, including a censored-data scenario. Real-data applications to waiting time and COVID-19 mortality datasets demonstrate competitive goodness of fit against both classical baselines and specialized APT variants, establishing the CAP-G framework as a flexible and tractable tool for reliability and survival modeling.