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Search Results (273)

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Keywords = Rényi entropy

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20 pages, 1074 KiB  
Article
A New Generalization of the Inverse Generalized Weibull Distribution with Different Methods of Estimation and Applications in Medicine and Engineering
by Ibtesam A. Alsaggaf, Sara F. Aloufi and Lamya A. Baharith
Symmetry 2024, 16(8), 1002; https://doi.org/10.3390/sym16081002 - 7 Aug 2024
Viewed by 729
Abstract
Limitations inherent to existing statistical distributions in capturing the complexities of real-world data often necessitate the development of novel models. This paper introduces the new exponential generalized inverse generalized Weibull (NEGIGW) distribution. The NEGIGW distribution boasts significant flexibility with symmetrical and asymmetrical shapes, [...] Read more.
Limitations inherent to existing statistical distributions in capturing the complexities of real-world data often necessitate the development of novel models. This paper introduces the new exponential generalized inverse generalized Weibull (NEGIGW) distribution. The NEGIGW distribution boasts significant flexibility with symmetrical and asymmetrical shapes, allowing its hazard rate function to be adapted to many failure patterns observed in various fields such as medicine, biology, and engineering. Some statistical properties of the NEGIGW distribution, such as moments, quantile function, and Renyi entropy, are studied. Three methods are used for parameter estimation, including maximum likelihood, maximum product of spacing, and percentile methods. The performance of the estimation methods is evaluated via Monte Carlo simulations. The NEGIGW distribution excels in its ability to fit real-world data accurately. Five medical and engineering datasets are applied to demonstrate the superior fit of NEGIGW distribution compared to competing models. This compelling evidence suggests that the NEGIGW distribution is promising for lifetime data analysis and reliability assessments across different disciplines. Full article
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32 pages, 414 KiB  
Article
Statistical Divergence and Paths Thereof to Socioeconomic Inequality and to Renewal Processes
by Iddo Eliazar
Entropy 2024, 26(7), 565; https://doi.org/10.3390/e26070565 - 30 Jun 2024
Viewed by 495
Abstract
This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The [...] Read more.
This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes. The general framework and its applications are shown to yield and to relate to the following: f-divergence, Hellinger divergence, Renyi divergence, and Kullback–Leibler divergence (also known as relative entropy); the Lorenz curve and socioeconomic inequality indices; the Gini index and its generalizations; the divergence of renewal processes from the Poisson process; and the divergence of anomalous relaxation from regular relaxation. Presenting a ‘fresh’ perspective on statistical divergence, this paper offers its readers a simple and transparent construction of statistical-divergence gauges, as well as novel paths that lead from statistical divergence to the aforementioned topics. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
28 pages, 2374 KiB  
Article
Convolutional Neural Networks for Local Component Number Estimation from Time–Frequency Distributions of Multicomponent Nonstationary Signals
by Vedran Jurdana and Sandi Baressi Šegota
Mathematics 2024, 12(11), 1661; https://doi.org/10.3390/math12111661 - 26 May 2024
Viewed by 556
Abstract
Frequency-modulated (FM) signals, prevalent across various applied disciplines, exhibit time-dependent frequencies and a multicomponent nature necessitating the utilization of time-frequency methods. Accurately determining the number of components in such signals is crucial for various applications reliant on this metric. However, this poses a [...] Read more.
Frequency-modulated (FM) signals, prevalent across various applied disciplines, exhibit time-dependent frequencies and a multicomponent nature necessitating the utilization of time-frequency methods. Accurately determining the number of components in such signals is crucial for various applications reliant on this metric. However, this poses a challenge, particularly amidst interfering components of varying amplitudes in noisy environments. While the localized Rényi entropy (LRE) method is effective for component counting, its accuracy significantly diminishes when analyzing signals with intersecting components, components that deviate from the time axis, and components with different amplitudes. This paper addresses these limitations and proposes a convolutional neural network-based (CNN) approach for determining the local number of components using a time–frequency distribution of a signal as input. A comprehensive training set comprising single and multicomponent linear and quadratic FM components with diverse time and frequency supports has been constructed, emphasizing special cases of noisy signals with intersecting components and differing amplitudes. The results demonstrate that the estimated component numbers outperform those obtained using the LRE method for considered noisy multicomponent synthetic signals. Furthermore, we validate the efficacy of the proposed CNN approach on real-world gravitational and electroencephalogram signals, underscoring its robustness and applicability across different signal types and conditions. Full article
(This article belongs to the Section Mathematics and Computer Science)
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14 pages, 701 KiB  
Article
Non-Hermitian Quantum Rényi Entropy Dynamics in Anyonic-PT Symmetric Systems
by Zhihang Liu and Chao Zheng
Symmetry 2024, 16(5), 584; https://doi.org/10.3390/sym16050584 - 9 May 2024
Viewed by 784
Abstract
We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable [...] Read more.
We reveal the continuous change of information dynamics patterns in anyonic-PT symmetric systems that originates from the continuity of anyonic-PT symmetry. We find there are three information dynamics patterns for anyonic-PT symmetric systems: damped oscillations with an overall decrease (increase) and asymptotically stable damped oscillations, which are three-fold degenerate and are distorted using the Hermitian quantum Rényi entropy or distinguishability. It is the normalization of the non-unitary evolved density matrix that causes the degeneracy and distortion. We give a justification for non-Hermitian quantum Rényi entropy being negative. By exploring the mathematics and physical meaning of the negative entropy in open quantum systems, we connect negative non-Hermitian quantum Rényi entropy and negative quantum conditional entropy, paving the way to rigorously investigate negative entropy in open quantum systems. Full article
(This article belongs to the Topic Quantum Information and Quantum Computing, 2nd Volume)
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16 pages, 18237 KiB  
Article
Novel Entropy for Enhanced Thermal Imaging and Uncertainty Quantification
by Hrach Ayunts, Artyom Grigoryan and Sos Agaian
Entropy 2024, 26(5), 374; https://doi.org/10.3390/e26050374 - 28 Apr 2024
Cited by 1 | Viewed by 1062
Abstract
This paper addresses the critical need for precise thermal modeling in electronics, where temperature significantly impacts system reliability. We emphasize the necessity of accurate temperature measurement and uncertainty quantification in thermal imaging, a vital tool across multiple industries. Current mathematical models and uncertainty [...] Read more.
This paper addresses the critical need for precise thermal modeling in electronics, where temperature significantly impacts system reliability. We emphasize the necessity of accurate temperature measurement and uncertainty quantification in thermal imaging, a vital tool across multiple industries. Current mathematical models and uncertainty measures, such as Rényi and Shannon entropies, are inadequate for the detailed informational content required in thermal images. Our work introduces a novel entropy that effectively captures the informational content of thermal images by combining local and global data, surpassing existing metrics. Validated by rigorous experimentation, this method enhances thermal images’ reliability and information preservation. We also present two enhancement frameworks that integrate an optimized genetic algorithm and image fusion techniques, improving image quality by reducing artifacts and enhancing contrast. These advancements offer significant contributions to thermal imaging and uncertainty quantification, with broad applications in various sectors. Full article
(This article belongs to the Special Issue Thermal Science and Engineering Applications)
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17 pages, 4576 KiB  
Article
Rotating Machinery Fault Diagnosis under Time–Varying Speed Conditions Based on Adaptive Identification of Order Structure
by Xinnan Yu, Xiaowang Chen, Minggang Du, Yang Yang and Zhipeng Feng
Processes 2024, 12(4), 752; https://doi.org/10.3390/pr12040752 - 8 Apr 2024
Viewed by 965
Abstract
Rotating machinery fault diagnosis is of key significance for ensuring safe and efficient operation of various industrial equipment. However, under nonstationary operating conditions, the fault–induced characteristic frequencies are often time–varying. Conventional Fourier spectrum analysis is not suitable for revealing time–varying details, and nonstationary [...] Read more.
Rotating machinery fault diagnosis is of key significance for ensuring safe and efficient operation of various industrial equipment. However, under nonstationary operating conditions, the fault–induced characteristic frequencies are often time–varying. Conventional Fourier spectrum analysis is not suitable for revealing time–varying details, and nonstationary fault feature extraction methods are still in desperate need. Order spectrum can reveal the rotational–speed–related time–varying frequency components as spectral peaks in order domain, thus facilitating fault feature extraction under time–varying speed conditions. However, the speed–unrelated frequency components are still nonstationary after angular–domain resampling, thus causing wide–band features and interferences in the order spectrum. To overcome such a drawback, this work proposes a rotating machinery fault diagnosis method based on adaptive separation of time–varying components and order feature extraction. Firstly, the rotational speed is estimated by the multi–order probabilistic approach (MOPA), thus eliminating the inconvenience of installing measurement equipment. Secondly, adaptive separation of the time–varying frequency component is achieved through time–varying filtering and surrogate test. It effectively eliminates interference from irrelevant components and noise. Finally, a high–resolution order spectrum is constructed based on the average amplitude envelope of each mono–component. It does not involve Fourier transform or angular–domain resampling, thus avoiding spectral leakage and resampling errors. By identifying the fault–related spectral peaks in the constructed order spectrum, accurate fault diagnosis can be achieved. The Rényi entropy values of the proposed order spectrum are significantly lower than those of the traditional order spectrum. This result verifies the effective energy concentration and high resolution of the proposed order spectrum. The results of both numerical simulation and lab experiments confirm the effectiveness of the proposed method in accurately presenting the time–varying frequency components for rotating machinery diagnosing faults. Full article
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14 pages, 907 KiB  
Article
New Generalized Weibull Inverse Gompertz Distribution: Properties and Applications
by Lamya A. Baharith
Symmetry 2024, 16(2), 197; https://doi.org/10.3390/sym16020197 - 7 Feb 2024
Cited by 1 | Viewed by 1001
Abstract
In parametric statistical modeling, it is essential to create generalizations of current statistical distributions that are more flexible when modeling actual data sets. Therefore, this study introduces a new generalized lifetime model named the odd Weibull Inverse Gompertz distribution (OWIG). The OWIG is [...] Read more.
In parametric statistical modeling, it is essential to create generalizations of current statistical distributions that are more flexible when modeling actual data sets. Therefore, this study introduces a new generalized lifetime model named the odd Weibull Inverse Gompertz distribution (OWIG). The OWIG is derived by combining the odd Weibull family of distributions with the inverse Gompertz distribution. Essential statistical properties are discussed, including reliability functions, moments, Rényi entropy, and order statistics. The proposed OWIG is particularly significant as its hazard rate functions exhibit various monotonic and nonmonotonic shapes. This enables OWIG to model different hazard behaviors more commonly observed in natural phenomena. OWIG’s parameters are estimated and its flexibility in predicting unique symmetric and asymmetric patterns is shown by analyzing real-world applications from psychology, environmental, and medical sciences. The results demonstrate that the proposed OWIG is an excellent candidate as it provides the most accurate fits to the data compared with some competing models. Full article
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10 pages, 1209 KiB  
Communication
Chirp Rate Estimation of LFM Signals Based on Second-Order Synchrosqueezing Transform
by Gangyi Zhai, Jianjiang Zhou, Kanglin Yu and Jiangtao Li
Electronics 2023, 12(24), 4938; https://doi.org/10.3390/electronics12244938 - 8 Dec 2023
Viewed by 881
Abstract
For the problem of low time-frequency aggregation of the short-time Fourier transform (STFT), which causes the parameter estimation performance degradation of linear frequency modulation (LFM) signals at low signal-to-noise ratio (SNR), second-order synchrosqueezing transform (SSST) is proposed based on the square of STFT [...] Read more.
For the problem of low time-frequency aggregation of the short-time Fourier transform (STFT), which causes the parameter estimation performance degradation of linear frequency modulation (LFM) signals at low signal-to-noise ratio (SNR), second-order synchrosqueezing transform (SSST) is proposed based on the square of STFT amplitude. The time-frequency resolution and energy aggregation are improved by means of squeezing and reassigning the time-frequency spectrum. Meanwhile, in order to decrease the calculation of classical parameter estimation methods, the Hough transform is used for rough estimation, and then the fractional Fourier transform (FRFT) is used for accuracy estimation based on the Renyi entropy. The simulation result shows that higher estimation accuracy is obtained at low SNR, and it has better robustness. Full article
(This article belongs to the Special Issue Advancements in Radar Signal Processing)
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23 pages, 4338 KiB  
Article
Probabilistic Procedures for SIR and SIS Epidemic Dynamics on Erdös-Rényi Contact Networks
by J. Leonel Rocha, Sónia Carvalho and Beatriz Coimbra
AppliedMath 2023, 3(4), 828-850; https://doi.org/10.3390/appliedmath3040045 - 16 Nov 2023
Viewed by 1154
Abstract
This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius [...] Read more.
This paper introduces the mathematical formalization of two probabilistic procedures for susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) infectious diseases epidemic models, over Erdös-Rényi contact networks. In our approach, we consider the epidemic threshold, for both models, defined by the inverse of the spectral radius of the associated adjacency matrices, which expresses the network topology. The epidemic threshold dynamics are analyzed, depending on the global dynamics of the network structure. The main contribution of this work is the relationship established between the epidemic threshold and the topological entropy of the Erdös-Rényi contact networks. In addition, a relationship between the basic reproduction number and the topological entropy is also stated. The trigger of the infectious state is studied, where the probability value of the stability of the infected state after the first instant, depending on the degree of the node in the seed set, is proven. Some numerical studies are included and illustrate the implementation of the probabilistic procedures introduced, complementing the discussion on the choice of the seed set. Full article
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28 pages, 804 KiB  
Article
Exploring the Entropy Complex Networks with Latent Interaction
by Alex Arturo Centeno Mejia and Moisés Felipe Bravo Gaete
Entropy 2023, 25(11), 1535; https://doi.org/10.3390/e25111535 - 11 Nov 2023
Viewed by 1154
Abstract
In the present work, we study the introduction of a latent interaction index, examining its impact on the formation and development of complex networks. This index takes into account both observed and unobserved heterogeneity per node in order to overcome the limitations of [...] Read more.
In the present work, we study the introduction of a latent interaction index, examining its impact on the formation and development of complex networks. This index takes into account both observed and unobserved heterogeneity per node in order to overcome the limitations of traditional compositional similarity indices, particularly when dealing with large networks comprising numerous nodes. In this way, it effectively captures specific information about participating nodes while mitigating estimation problems based on network structures. Furthermore, we develop a Shannon-type entropy function to characterize the density of networks and establish optimal bounds for this estimation by leveraging the network topology. Additionally, we demonstrate some asymptotic properties of pointwise estimation using this function. Through this approach, we analyze the compositional structural dynamics, providing valuable insights into the complex interactions within the network. Our proposed method offers a promising tool for studying and understanding the intricate relationships within complex networks and their implications under parameter specification. We perform simulations and comparisons with the formation of Erdös–Rényi and Barabási–Alber-type networks and Erdös–Rényi and Shannon-type entropy. Finally, we apply our models to the detection of microbial communities. Full article
(This article belongs to the Section Complexity)
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30 pages, 692 KiB  
Article
Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes
by Anatoliy Malyarenko, Yuliya Mishura, Kostiantyn Ralchenko and Yevheniia Anastasiia Rudyk
Axioms 2023, 12(11), 1026; https://doi.org/10.3390/axioms12111026 - 31 Oct 2023
Cited by 2 | Viewed by 1065
Abstract
We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian [...] Read more.
We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes. Full article
(This article belongs to the Special Issue Stochastic Processes in Quantum Mechanics and Classical Physics)
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12 pages, 2128 KiB  
Article
Pairing Superfluid–Insulator Transition Induced by Atom–Molecule Conversion in Bosonic Mixtures in Optical Lattice
by Haiming Deng, Zhi Tan, Chao Kong, Fuqiu Ye and Honghua Zhong
Symmetry 2023, 15(9), 1715; https://doi.org/10.3390/sym15091715 - 7 Sep 2023
Viewed by 1089
Abstract
Motivated by the recent experiment on bosonic mixtures of atoms and molecules, we investigate pairing superfluid–insulator (SI) transition for bosonic mixtures of atoms and molecules in a one-dimensional optical lattice, which is described by an extended Bose–Hubbard model with atom–molecule conservation (AMC). It [...] Read more.
Motivated by the recent experiment on bosonic mixtures of atoms and molecules, we investigate pairing superfluid–insulator (SI) transition for bosonic mixtures of atoms and molecules in a one-dimensional optical lattice, which is described by an extended Bose–Hubbard model with atom–molecule conservation (AMC). It is found that AMC can induce an extra pair–superfluid phase though the system does not demonstrate pair-hopping. In particular, the system may undergo several pairing SI or insulator–superfluid transitions as the detuning from the Feshbach resonance is varied from negative to positive, and the larger positive detuning can bifurcate the pair–superfluid phases into mixed superfluid phases consisting of single-atomic and pair-atomic superfluid. The calculation of the second-order Rényi entropy reveals that the discontinuity in its first-order derivative corresponds to the phase boundary of the pairing SI transition. This means that the residual entanglement in our mean-field treatment can be used to efficiently capture the signature of the pairing SI transition induced by AMC. Full article
(This article belongs to the Section Physics)
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12 pages, 304 KiB  
Article
Fractal Divergences of Generalized Jacobi Polynomials
by Răzvan-Cornel Sfetcu and Vasile Preda
Mathematics 2023, 11(16), 3500; https://doi.org/10.3390/math11163500 - 13 Aug 2023
Viewed by 883
Abstract
The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal [...] Read more.
The notion of entropy (including macro state entropy and information entropy) is used, among others, to define the fractal dimension. Rényi entropy constitutes the basis for the generalized correlation dimension of multifractals. A motivation for the study of the information measures of orthogonal polynomials is because these polynomials appear in the densities of many quantum mechanical systems with shape-invariant potentials (e.g., the harmonic oscillator and the hydrogenic systems). With the help of a sequence of some generalized Jacobi polynomials, we define a sequence of discrete probability distributions. We introduce fractal Kullback–Leibler divergence, fractal Tsallis divergence, and fractal Rényi divergence between every element of the sequence of probability distributions introduced above and the element of the equiprobability distribution corresponding to the same index. Practically, we obtain three sequences of fractal divergences and show that the first two are convergent and the last is divergent. Full article
(This article belongs to the Section Difference and Differential Equations)
19 pages, 3359 KiB  
Article
Asymmetric Right-Skewed Size-Biased Bilal Distribution with Mathematical Properties, Reliability Analysis, Inference and Applications
by Amer Ibrahim Al-Omari, Rehab Alsultan and Ghadah Alomani
Symmetry 2023, 15(8), 1578; https://doi.org/10.3390/sym15081578 - 13 Aug 2023
Viewed by 918
Abstract
Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) [...] Read more.
Asymmetric distributions, as opposed to symmetric distributions, may be more resilient to extreme values or outliers. Furthermore, when data show substantial skewness, asymmetric distributions can shed light on the underlying processes or phenomena being investigated. In this direction, the size-biased Bilal distribution (SBBD) is suggested in this study as a generalization to the Bilal distribution. The length-biased and area-biased Bilal distributions are discussed in detail as two special cases. The main statistical properties of the distribution including the rth moment, coefficients of variation, skewness, kurtosis, moment generating function, incomplete moments, moments of residual life, harmonic mean, Fisher’s information, and the Rényi entropy as a measure of uncertainty are presented. Graphical representations of the cumulative distribution, probability density, odds, survival, hazard, reversed hazard rate, and cumulative hazard functions are presented for further explanation of the distribution behavior. In addition, the methods of moments and maximum likelihood estimates are taken into account for estimating the model parameters. A simulation study is carried out to see the efficiency of the maximum likelihood in terms of standard errors and bias. Real data sets of precipitation and myeloid leukemia patients are considered to show the practical significance of the suggested distributions as an alternative to some well-known distributions such as the Rama, Rani, Bilal, and exponential distributions. It is found that the size-biased Bilal distribution is right-skewed and has a superior fitting performance compared to the other distributions in this study. Full article
(This article belongs to the Special Issue Symmetry in Probability Theory and Statistics)
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29 pages, 1122 KiB  
Article
The New Exponentiated Half Logistic-Harris-G Family of Distributions with Actuarial Measures and Applications
by Gayan Warahena-Liyanage, Broderick Oluyede, Thatayaone Moakofi and Whatmore Sengweni
Stats 2023, 6(3), 773-801; https://doi.org/10.3390/stats6030050 - 31 Jul 2023
Cited by 3 | Viewed by 1176
Abstract
In this study, we introduce a new generalized family of distributions called the Exponentiated Half Logistic-Harris-G (EHL-Harris-G) distribution, which extends the Harris-G distribution. The motivation for introducing this generalized family of distributions lies in its ability to overcome the limitations of previous families, [...] Read more.
In this study, we introduce a new generalized family of distributions called the Exponentiated Half Logistic-Harris-G (EHL-Harris-G) distribution, which extends the Harris-G distribution. The motivation for introducing this generalized family of distributions lies in its ability to overcome the limitations of previous families, enhance flexibility, improve tail behavior, provide better statistical properties and find applications in several fields. Several statistical properties, including hazard rate function, quantile function, moments, moments of residual life, distribution of the order statistics and Rényi entropy are discussed. Risk measures, such as value at risk, tail value at risk, tail variance and tail variance premium, are also derived and studied. To estimate the parameters of the EHL-Harris-G family of distributions, the following six different estimation approaches are used: maximum likelihood (MLE), least-squares (LS), weighted least-squares (WLS), maximum product spacing (MPS), Cramér–von Mises (CVM), and Anderson–Darling (AD). The Monte Carlo simulation results for EHL-Harris-Weibull (EHL-Harris-W) show that the MLE method allows us to obtain better estimates, followed by WLS and then AD. Finally, we show that the EHL-Harris-W distribution is superior to some other equi-parameter non-nested models in the literature, by fitting it to two real-life data sets from different disciplines. Full article
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