Unit Shiha Distribution and its Applications to Engineering and Medical Data

TL;DR

The paper addresses the need for flexible models for data on the unit interval by introducing the Unit Shiha (USh) distribution, obtained via the inverse exponential transform $X=e^{-Y}$ with $Y\sim Shiha(\omega,\eta)$. It derives the PDF and CDF, analyzes hazard-rate shapes, and develops key properties including moments, a numerical quantile function, entropy, and stress-strength reliability (noting $R_{USh}=1-R_{Sh}$). Parameter estimation is performed via maximum likelihood, with a comprehensive simulation study confirming good finite-sample behavior and coverage, and the method is validated on four real data sets where USh outperforms competing unit distributions according to multiple criteria and diagnostic plots. The results demonstrate the practical usefulness of the USh distribution for modeling diverse unit-valued data in engineering and medical contexts, offering a flexible, well-supported alternative in reliability and bounded-data analysis.

Abstract

There is a growing need for flexible statistical distributions that can accurately model data defined on the unit interval. This paper introduces a new unit distribution, termed the unit Shiha (USh) distribution, which is derived from the original Shiha (Sh) distribution through an inverse exponential transformation. The probability density function of the USh distribution is sufficiently flexible to model both left- and right-skewed data, while its hazard rate function is capable of capturing various failure-rate patterns, including increasing, bathtub-shaped, and J-shaped forms. Several statistical properties of the proposed distribution are investigated, including moments and related measures, the quantile function, entropy, and stress-strength reliability. Parameter estimation is carried out using the maximum likelihood method, and its performance is evaluated through a simulation study. The practical usefulness of the USh distribution is demonstrated using four real-life data sets, and its performance is compared with several well-known competing unit distributions. The comparative results indicate that the proposed model fits the data better than the competitive models applied in this study.

Figures (20)

• Figure 1: The PDF and hazard functions of the UShD at different parameter values.
• Figure 2: The variance, skewness, and kurtosis of the UShD for $0< \omega, \eta \leq 10$.
• Figure 3: Left: quantile functions of UShD for $\eta = 0.4$ and various $\omega$ values. Right: quartiles of UShD as functions of $\omega$ for $\eta = 0.4$.
• Figure 4: The entropy of UShD for $0< \omega, \eta \leq 12$, the entropy range is [-3.576, -0.003]
• Figure 5: TTT, box, and violin plots for data I.

Theorems & Definitions (2)

• Theorem 1
• proof