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There are some generalizations of the classical exponential distribution in the statistical literature that have proven to be helpful in numerous scenarios. Some of these distributions are the families of distributions that were proposed by Marshall and Olkin and Gupta. The disadvantage of these models is the impossibility of fitting data of a bimo...
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... this case, the parameter θ can be interpreted as a precision parameter, since the function m(θ) increases for −3 < θ < 1 (see Figure 3) and decreases in the rest of the domain of θ. Thus, µ is the mean of the response variable and θ can be regarded as a precision parameter in the sense that, for a fixed value of µ, the variance of Y varies according to the values of m(θ), i.e., the values of the parameter θ. ...
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... Researchers can explore various aspects of risk analysis, such as dependence modeling, copulas, multivariate extensions, and time series modeling, using the Lomax distribution as a building block. For more details, regression models, applications, and real datasets, see Butt and Khalil [14] for a novel skewed bimodal model for modeling the asymmetric heavy-tail bimodal datasets, Reyes et al. [15] for a new bimodal exponential extension with some applications in risk theory, and Gómez et al. [16] for asymmetric bimodal double-regression modeling. A few plots of the GELX PDF and HRF are shown in Figure 1. ...
The idea of symmetry, which is used to describe the shape of a probability distribution, is a key concept in the theory of probability. The use of symmetric and asymmetric distributions is common in statistical inference, decision-making, and probability calculations. This article introduces a novel asymmetric model for assessing risks under a skewed claims dataset. The new distribution is also employed for both censored and uncensored validation testing. Four estimation methods, maximum likelihood, ordinary least squares, L-Moment, and Anderson Darling, were used for the risk assessment and analysis. To explain the exposure to risk within actuarial claims data, we introduced five crucial indicators, namely value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess losses. A numerical and graphical analysis is presented to assess the actuarial risk. Furthermore, the article discusses a newly developed Rao Robson Nikulin statistic for censored and uncensored validation testing. The validation testing also involved the insurance claims dataset.
... The probability density function of this distribution writes as follow: [8,11,15] In fact, the gamma distribution is derived from the gamma function, or what is occasionally named gamma integral, which is mentioned in many advanced mathematics books. This function of distribution can be considered as one of the major distributions employed to studying problems in which time is one of its factors, such as those specialized studies on the length of the operating time of equipment the particular factory. ...
In this paper, we attempt to diagnosis of some of exponential distributions, represent by the normal distribution as a criteria with two others distributions (Gamma & Exponential) distributions we attempt to analyze the effect of the number of parameter(s) two or one parameter(s) in the distribution on the maximize likelihood estimation for the three distributions. That proved the maximum likelihood estimates (MLE’s) of the parameters; it's obtained by using some packages (e.g. fitdistrplus) insides some algorithms in (R) that based on behavior of the simulated data like parameter and sample size. A comparison is carried out between the mentioned distributions based on the classical kolmogrov-smirnov distance to minimize distance estimation test statistic and Grammer von misses distance, to emphasize that which distribution fitted to the data better than the other models. After completing this, A good preface to some distributions is to summarize the relationships between observations. How to calculate and Plot Probability and Density Functions the normal distribution. Keywords: Maximization of likelihood estimations (MLE), exponential family, density functions, fitdistrplus package, Grammer von misses distance.
... The LBeta-G family [1], Kumaraswamy-G family [2], Topp-Leone G family [3], modified beta transmuted-G family [4], exponentiated Weibull distribution [5], Odd-Lidney half-logistic distribution [6], and extended Gumbel distribution [7] are examples of these types of derived distributions. Among others, the weighted exponential [8], Nadarajah-Haghighi exponential [9], exponentiated generalized linear exponential [10], exponentiated Weibull (EW) [11], exponentiated Weibull-Poisson (EWP) [12], extended exponential [13], α-power transformed generalized exponential [14], odd exponentiated halflogistic exponential [15], exponentiated additive Weibull [16], exponentiated Weibull-exponential [17], extended odd Weibull exponential (EOWEx) [18], and bimodal exponential [19] distributions are extensions of the exponential distribution frequently used in survival analysis. Exponentiated distributions are obtained by exponentiating existing distributions. ...
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... This dataset was obtained from anonymized data published by the Israeli Ministry of Health and it was analyzed in a previous work [47]. The dataset contains the following values: 16,16,16,14,36,9,10,11,8,9,12,10,22,5,11,17,20,12,29,12,15,25,25,24,18,13,44,14,20,19,11,10,18,21,31,9,29,12,10,10,13,12,19,33,37,16,63,9,28, and 16. We analyze this dataset to compare the GMO-EE with the W, EE [24], MOEBXII [41], GBE-II [42], EOWEx [18], EWP [12], and exponential distributions. ...
In this study, we propose a generalized Marshall-Olkin exponentiated exponential distribution as a submodel of the family of generalized Marshall-Olkin distribution. Some statistical properties of the proposed distribution are examined such as moments, the moment-generating function, incomplete moment, and Lorenz and Bonferroni curves. We give five estimators for the unknown parameters of the proposed distribution based on maximum likelihood, least squares, weighted least squares, and the Anderson-Darling and Cramer-von Mises methods of estimation. To investigate the finite sample properties of the estimators, a comprehensive Monte Carlo simulation study is conducted for the models with three sets of randomly selected parameter values. Finally, four different real data applications are presented to demonstrate the usefulness of the proposed distribution in real life.
... Other practical examples of bimodality in data can be seen in [4][5][6]. In the literature, there are many proposals discussing bimodal distributions; e.g., the works of [7][8][9][10][11][12]. Bimodal data can be fitted by a mixture of two unimodal distributions. ...
In this paper, we introduce an extension of the sinh Cauchy distribution including a double regression model for both the quantile and scale parameters. This model can assume different shapes: unimodal or bimodal, symmetric or asymmetric. We discuss some properties of the model and perform a simulation study in order to assess the performance of the maximum likelihood estimators in finite samples. A real data application is also presented.
... In the field of security, the method of fuzzy logic (fuzzy model) is well known for its probabilistic character, which can be applied to estimate the future development of risks [10][11][12][13][14][15][16]. Another important method is game theory and its application in various studies examining the level of security or defining various scenarios of the development of the threat of potential risks using e.g., Bayesian functions [17][18][19][20], or other statistical distribution and exponential distribution in the planned security risk scenario [21]. Risk assessment is applicable across the whole spectrum of security objects, whether there are threats to physical objects [22][23][24], information systems, or other financial systems [25]. ...
The existence or non-existence of a threat to a system is essential for its existence or essential for the functionality of the system. Even more crucial is the potential of the threat and its development, which leads to the failure of the symmetry of the system. What influences the development of such threats? What contexts influence the evolution of system threats? The development of threats is linked to the changing values of indicators that affect the state of the threat at a certain point in time. This development takes place in a constantly changing environment, therefore it is dynamically and causally linked. The system aims to maintain its order, however, the influence of the development of threats deflects it towards the entropy of the system. The paper is focused on the identification of the phases of the development of threats and their impact on the symmetry of a system. The paper presents a theoretical view of the impact of threat development on system symmetry failure.
