A new extension of generalized beta distribution: statistical properties, prediction estimation and application in failure components and physics data

This study presents a generalized function of the beta distribution combined with the gamma distribution, resulting in the compound generalized beta (CG-Beta) distribution. The distribution's attributes examined include the probability density function, cumulative distribution function, reliability function, and hazard function. Essential statistical qualities include the median, quantile function, inverted hazard rate function, Mills Ratio, cumulative risk function, probability weighted moments, moments, Rényi entropy, ω-entropy, and generalized order statistics. We examine the estimation of model parameters, reliability, and hazard rate functions of the CG-Beta distribution utilizing Bayesian estimation and maximum likelihood approaches. The appropriate estimated confidence intervals of unknown parameters are likewise built using the probability estimators' normal approximations. The Bayesian estimators are created using independent gamma conjugate priors under the symmetrical squared-error loss function. The Bayesian estimators and corresponding highest posterior density intervals cannot be estimated analytically due to the complex shape of the joint likelihood function; nevertheless, they can be evaluated using Monte Carlo Markov chain (MCMC) methods. Comprehensive Monte Carlo simulations assess the proposed estimation approaches. Additionally, the maximum likelihood and Bayesian predictions of future observations are shown. A simulated study assesses the efficacy of the suggested distribution. Ultimately, two datasets are examined under the proposed distributions to demonstrate their adaptability for practical use.