Unit compound Rayleigh model: Statistical characteristics, estimation and application

Qin Gong; Laijun Luo; Haiping Ren; Qin Gong; Laijun Luo; Haiping Ren

doi:10.3934/math.20241110

AIMS Mathematics

2024, Volume 9, Issue 8: 22813-22841. doi: 10.3934/math.20241110

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Unit compound Rayleigh model: Statistical characteristics, estimation and application

1.
College of Science, Jiangxi University of Science and Technology, Ganzhou, 341000, China
2.
School of Software Engineering, Jiangxi University of Science and Technology, Nanchang, 330013, China

Received: 16 May 2024 Revised: 09 July 2024 Accepted: 18 July 2024 Published: 23 July 2024
MSC : 62E10, 62F10

In this paper, we proposed a novel probability distribution model known as the unit compound Rayleigh distribution, which possesses the distinctive characteristic of defining the range within the bounded interval (0, 1). Through an in-depth investigation of this distribution, we analyzed various statistical and structural characteristics including reliability function, risk function, quantile function, moment analysis, order statistics, and entropy measurement. To estimate the unknown parameters of our proposed distribution model, we employed maximum likelihood (ML) estimation and Bayesian estimation. Furthermore, we derived several entropy measures based on ML estimation under the unit compound Rayleigh distribution. To comprehensively evaluate the performance of these entropies, we employed the Monte Carlo simulation method to calculate the average entropy estimate, average entropy bias, corresponding mean square error, and mean relative estimate for assessing the performance of various entropies within the unit compound Rayleigh distribution model. Finally, in order to validate its potential for practical applications, two sets of real data were selected for empirical analysis where fitting and parameter estimation were conducted to demonstrate the advantages of utilizing the unit compound Rayleigh distribution in describing and predicting actual data. This study not only introduces a new probability theory and statistics framework by proposing a novel distribution model but also provides researchers and practitioners in related fields with a powerful analytical tool.
- unit compound Rayleigh distribution,
- statistical characteristics,
- maximum likelihood estimation,
- Bayesian estimation,
- Shannon entropy,
- Lorentz curve
Citation: Qin Gong, Laijun Luo, Haiping Ren. Unit compound Rayleigh model: Statistical characteristics, estimation and application[J]. AIMS Mathematics, 2024, 9(8): 22813-22841. doi: 10.3934/math.20241110

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Abstract

In this paper, we proposed a novel probability distribution model known as the unit compound Rayleigh distribution, which possesses the distinctive characteristic of defining the range within the bounded interval (0, 1). Through an in-depth investigation of this distribution, we analyzed various statistical and structural characteristics including reliability function, risk function, quantile function, moment analysis, order statistics, and entropy measurement. To estimate the unknown parameters of our proposed distribution model, we employed maximum likelihood (ML) estimation and Bayesian estimation. Furthermore, we derived several entropy measures based on ML estimation under the unit compound Rayleigh distribution. To comprehensively evaluate the performance of these entropies, we employed the Monte Carlo simulation method to calculate the average entropy estimate, average entropy bias, corresponding mean square error, and mean relative estimate for assessing the performance of various entropies within the unit compound Rayleigh distribution model. Finally, in order to validate its potential for practical applications, two sets of real data were selected for empirical analysis where fitting and parameter estimation were conducted to demonstrate the advantages of utilizing the unit compound Rayleigh distribution in describing and predicting actual data. This study not only introduces a new probability theory and statistics framework by proposing a novel distribution model but also provides researchers and practitioners in related fields with a powerful analytical tool.

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