A New Flexible Transmuted Distribution: Theory and Application

Abstract

In this paper, we introduce a new three-parameter lifetime distribution, termed the Transmuted Ibrahim (TI) distribution, which is constructed by applying the quadratic rank transmutation map to the Ibrahim distribution. The TI distribution represents a novel three-parameter lifetime model arising from the quadratic rank transmutation of the Ibrahim distribution. In contrast to many existing transmuted families that emphasize purely formal generalization, the TI distribution is specifically designed to overcome concrete shortcomings of the original Ibrahim model—particularly its poor performance in representing high kurtosis and pronounced right-tail behavior—while maintaining analytical and computational feasibility. The proposed framework addresses key weaknesses of the baseline Ibrahim distribution, most notably its inability to adequately model heavy tails and pronounced asymmetry that often appear in reliability and survival data.We derive a broad set of statistical properties, including closed-form expressions for ordinary and incomplete moments, quantile functions, generating functions, and Rényi entropy. Conditions for unimodality are established, and the corresponding modal behavior is analyzed. Parameter inference is carried out using maximum likelihood estimation and the method of moments. A detailed Monte Carlo simulation study—implemented with careful attention to numerical robustness and convergence behavior—shows that the maximum likelihood estimators are consistent over a range of sample sizes.The practical relevance of the TI model is demonstrated by fitting it to two real datasets: guinea pig and rat survival times. Its performance is benchmarked against the Weibull, Gamma, and Generalized Exponential distributions. For the guinea pig data, the TI distribution yields the best fit, as indicated by the smallest AIC value (787.424) and the largest Kolmogorov–Smirnov (KS) p-value (0.5316). For the rat survival data, the TI model attains the lowest KS statistic (0.160) and the highest p-value (0.1363) among all competing models. We also provide a critical assessment of the TI distribution’s limitations, including potential identifiability challenges near boundary parameter values, and we propose several promising directions for subsequent research.