article · Computational Journal of Mathematical and statistical Sciences
A new statistical framework, termed the doubly truncated Zubair-generalized family of distributions, has been developed alongside an analysis of its mathematical properties. These properties include quantiles, central and non-central moments, order statistics, and entropy measures such as Rényi, Shannon, and Tsallis entropies. Measures relating to lifetime analysis, including mean residual life, mean past lifetime, and mean time to failure, were also examined. A specific sub-model, the doubly truncated Zubair-Weibull distribution, was evaluated for reliability, hazard, reversed hazard, and cumulative hazard rate functions. Parameter estimation for this model was carried out using the maximum likelihood estimation approach, with its performance verified through a simulation study. Finally, the model demonstrated its flexibility and practical applicability when tested against two real-world lifetime datasets.
Real-world lifetime and failure phenomena often fall within bounded intervals where standard distributions struggle to fit accurately. Providing a truncated probability distribution family equipped with flexible hazard and reliability functions enables researchers and practitioners to model bounded survival and failure data more effectively, improving statistical accuracy in reliability and survival analyses.
The model could assist data analysts and reliability engineers who require accurate lifetime and failure rate modelling for bounded systems. Because the research is confined to theoretical derivations, simulations, and fitting to two real lifetime datasets, it remains early-stage research. Commercial utility would depend on incorporating the distribution into existing statistical software libraries for industrial reliability assessments.
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This paper proposed a truncated family of probability distributions named the doubly truncated Zubair-generalized family of truncated distributions. Certain properties of doubly truncated Zubair-Generalized family of truncated distributions are worked on. These properties are the quantile, median, the non-central moments, central moments, order statistics, entropy measures such as R$\acute{e}$nyi, Shannon, Tsallis entropies, also the mean residual life, mean past lifetime and mean time to failure. The doubly truncated Zubair-Weibull distribution is a special sub-model of the doubly truncated Zubair-generalized family. Some important statistical properties of the doubly truncated Zubair-Weibull distribution are studied, such as the reliability, hazard, reversed hazard and cumulative hazard rate functions. In addition, the moments, quantile, order statistics, entropies and some important special sub-models of the doubly truncated Zubair-Weibull distribution are given. The maximum likelihood estimation approach is applied to estimate the unknown parameters, reliability and hazard rate functions. A simulation study is conducted to evaluate the performance of the maximum likelihood estimates. Two life-time real data sets are applied to show the flexibility and applicability of the proposed model.
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DOI: 10.21608/cjmss.2024.322714.1073
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