Quartic Rank Transmutation of Gamma–Type Distributions: Characteristics and Estimation

Abstract

Rank transmutation maps have emerged as one of the adopted methods for proposing new probability distributions. This study used the quartic rank transmutation to introduce three new probability distributions: the Quartic Transmuted Exponential Distribution, Quartic Transmuted Lindley Distribution, and Quartic Transmuted Rayleigh Distribution. The construction of these distributions involves meticulously examining mathematical concepts, encompassing probability density functions, survival functions, moments, entropies, and order statistics. Visual aids, including cumulative distribution functions, probability density functions, and hazard rate functions, enhance the comprehension of distribution characteristics. A comprehensive simulation study underscores a consistent trend: a reduction in bias for maximum likelihood estimation and refinement in standard errors with increasing sample size. The practical applicability of these newly proposed distributions was demonstrated using real-world datasets. The quartic transmuted exponential distribution was effectively employed to model the lifetime of 50 devices, referencing data from Aarset’s study in 1987. Similarly, the quartic transmuted Lindley distribution was adeptly applied to remission times (measured in months) of 128 bladder cancer patients. Finally, the quartic transmuted Rayleigh distribution was successfully utilized to analyze a dataset comprising 72 instances of exceedance from the Wheaton River flood data near Carcoss in Yukon Territory, Canada. Evaluation criteria such as log-likelihood, AIC, AICc, and BIC affirm the superior flexibility and performance of the proposed distributions. This research significantly contributes to distribution theory, offering innovative methods to enhance distribution adaptability in diverse applications.