Shiha Distribution: Statistical Properties and Applications to Reliability Engineering and Environmental Data
F. A. Shiha
TL;DR
We introduce Shiha, a flexible two-parameter lifetime distribution defined as a mixture of Exp$(\omega)$, Exp$(2\omega)$, and Gamma$(2,2\omega)$ with weights $p_1=\frac{\omega}{\omega+3\eta}$, $p_2=\frac{\eta}{\omega+3\eta}$, $p_3=\frac{2\eta}{\omega+3\eta}$. Key analytic results include the pdf, cdf, survival, hazard rate, quantile function (requiring numerical solution via $F(y_p;\omega,\eta)=p$), moment generating function $M_Y(t)=\frac{4\omega^{4}+12\omega^{3}\eta-4\omega^{3}t-14\omega^{2}\eta t+\omega^{2}t^{2}+2\omega\eta t^{2}}{(\omega+3\eta)(\omega-t)(2\omega-t)^{2}}$ for $t<\omega$, and entropy $H(y)=-\int_0^{\infty} f(y;\omega,\eta)\ln f(y;\omega,\eta)\,dy$ (computed numerically). The stress--strength reliability $R=\int_0^{\infty} f(y;\omega_1,\eta_1) F(y;\omega_2,\eta_2)\,dy$ has closed-form evaluations and reduces correctly to standard Exp–Exp cases when $\eta=0$. A Monte Carlo study confirms that the MLEs $\hat{\omega},\hat{\eta}$ are consistent as sample size grows, and applications to four real datasets show Shiha consistently provides superior fits relative to several competing lifetime models, as evidenced by lower AIC/BIC and higher AD/KS p-values. Overall, Shiha offers flexible hazard-rate shapes and tail behavior, making it suitable for reliability engineering and environmental data analysis.
Abstract
This paper introduces a new two-parameter distribution, referred to as the Shiha distribution, which provides a flexible model for skewed lifetime data with either heavy or light tails. The proposed distribution is applicable to various fields, including reliability engineering, environmental studies, and related areas. We derive its main statistical properties, including the moment generating function, moments, hazard rate function, quantile function, and entropy. The stress--strength reliability parameter is also derived in closed form. A simulation study is conducted to evaluate its performance. Applications to several real data sets demonstrate that the Shiha distribution consistently provides a superior fit compared with established competing models, confirming its practical effectiveness for lifetime data analysis.