A Flexible Truncated (<i>u</i>,<i>v</i>)-Half-Normal Distribution: Properties, Estimation and Applications

A Flexible Truncated (<i>u</i>,<i>v</i>)-Half-Normal Distribution: Properties, Estimation and Applications

This study introduces the truncated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow&g...

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Bibliographic Details
Main Authors:	Maher Kachour, Hassan S. Bakouch, Mustapha Muhammad, Badamasi Abba, Lamia Alyami, Sadiah M. A. Aljeddani
Format:	Article
Language:	English
Published:	MDPI AG 2025-05-01
Series:	Mathematics
Subjects:	half normal distribution moments simulation parameter estimation sensitivity analysis testing algorithm
Online Access:	https://www.mdpi.com/2227-7390/13/11/1740
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Summary:	This study introduces the truncated <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>-half-normal distribution, a novel probability model defined on the bounded interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow></semantics></math></inline-formula>, with parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> and <i>b</i>. This distribution is designed to model processes with restricted domains, ensuring realistic and analytically tractable outcomes. Some key properties of the proposed model, including its cumulative distribution function, probability density function, survival function, hazard rate, and moments, are derived and analyzed. Parameter estimation of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> and <i>b</i> is achieved through a hybrid approach, combining maximum likelihood estimation (MLE) for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> and a likelihood-free-inspired technique for <i>b</i>. A sensitivity analysis highlighting the dependence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> on <i>b</i>, and an optimal estimation algorithm is proposed. The proposed model is applied to two real-world data sets, where it demonstrates superior performance over some existing models based on goodness-of-fit criteria, such as the known AIC, BIC, CAIC, KS, AD, and CvM statistics. The results emphasize the model’s flexibility and robustness for practical applications in modeling data with bounded support.
ISSN:	2227-7390