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Journal articles on the topic 'Exponential Family of distribution'

Author: Grafiati

Published: 28 June 2021

Last updated: 1 February 2022

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1

Mahmoud, Mahmoud Riad, Moshera A. M. Ahmad, and AzzaE Ismail. "T-Inverse Exponential Family Of Distributions." Journal of University of Shanghai for Science and Technology 23, no. 09 (September 13, 2021): 556–72. http://dx.doi.org/10.51201/jusst/21/08495.

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Recently, several methods have been introduced to generate neoteric distributions with more exibility, like T-X, T-R [Y] and alpha power. The T-Inverse exponential [Y] neoteric family of distributons is proposed in this paper utilising the T-R [Y] method. A generalised inverse exponential (IE) distribution family has been established. The distribution family is generated using quantile functions of some dierent distributions. A number of general features in the T-IE [Y] family are examined, like mean deviation, mode, moments, quantile function, and entropies. A special model of the T-IE [Y] distribution family was one of those old distributions. Certain distribution examples are produced by the T-IE [Y] family. An applied case was presented which showed the importance of the neoteric family.

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Makubate, Boikanyo, Broderick O. Oluyede, Gofaone Motobetso, Shujiao Huang, and Adeniyi F. Fagbamigbe. "The Beta Weibull-G Family of Distributions: Model, Properties and Application." International Journal of Statistics and Probability 7, no. 2 (January 18, 2018): 12. http://dx.doi.org/10.5539/ijsp.v7n2p12.

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A new family of generalized distributions called the beta Weibull-G (BWG) distribution is proposed and developed. This new class of distributions has several new and well known distributions including exponentiated-G, Weibull-G, Rayleigh-G, exponential-G, beta exponential-G, beta Rayleigh-G, beta Rayleigh exponential, beta-exponential-exponential, Weibull-log-logistic distributions, as well as several other distributions such as beta Weibull-Uniform, beta Rayleigh-Uniform, beta exponential-Uniform, beta Weibull-log logistic and beta Weibull-exponential distributions as special cases. Series expansion of the density function, hazard function, moments, mean deviations, Lorenz and Bonferroni curves, R\'enyi entropy, distribution of order statistics and maximum likelihood estimates of the model parameters are given. Application of the model to real data set is presented to illustrate the importance and usefulness of the special case beta Weibull-log-logistic distribution.

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Block, Henry W., Naftali A. Langberg, and Thomas H. Savits. "A MIXTURE OF EXPONENTIAL AND IFR GAMMA DISTRIBUTIONS HAVING AN UPSIDEDOWN BATHTUB-SHAPED FAILURE RATE." Probability in the Engineering and Informational Sciences 26, no. 4 (July 30, 2012): 573–80. http://dx.doi.org/10.1017/s0269964812000204.

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We consider a mixture of one exponential distribution and one gamma distribution with increasing failure rate. For the right choice of parameters, it is shown that its failure rate has an upsidedown bathtub shape failure rate. We also consider a mixture of a family of exponentials and a family of gamma distributions and obtain a similar result.

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Louzada, Francisco, Vitor Marchi, and James Carpenter. "The Complementary Exponentiated Exponential Geometric Lifetime Distribution." Journal of Probability and Statistics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/502159.

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We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its survival and hazard functions, moments,rth moment of theith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution.

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Ghorbanpour, Samereh, Rahim Chinipardaz, and Seyed Mohammad Reza Alavi. "Form-Invariance of the Non-Regular Exponential Family of Distributions." Revista Colombiana de Estadística 41, no. 2 (July 1, 2018): 157–72. http://dx.doi.org/10.15446/rce.v41n2.62233.

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The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the sufficient and minimal sufficient statistics, maximum likelihood estimation and the Fisher information matrix are studied.

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Iwasaki, Masakazu, and Hiroe Tsubaki. "A new bivariate distribution in natural exponential family." Metrika 61, no. 3 (June 2005): 323–36. http://dx.doi.org/10.1007/s001840400348.

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Abdulkadir, Dr Sauta Saidu, J. Jerry, and T. G. Ieren. "STATISTICAL PROPERTIES OF LOMAX-INVERSE EXPONENTIAL DISTRIBUTION AND APPLICATIONS TO REAL LIFE DATA." FUDMA JOURNAL OF SCIENCES 4, no. 2 (October 7, 2020): 680–94. http://dx.doi.org/10.33003/fjs-2020-0402-435.

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This paper proposes a Lomax-inverse exponential distribution as an improvement on the inverse exponential distribution in the form of Lomax-inverse Exponential using the Lomax generator (Lomax-G family) with two extra parameters to generalize any continuous distribution (CDF). The probability density function (PDF) and cumulative distribution function (CDF) of the Lomax-inverse exponential distribution are defined. Some basic properties of the new distribution are derived and extensively studied. The unknown parameters estimation of the distribution is done by method of maximum likelihood estimation. Three real-life datasets are used to assess the performance of the proposed probability distribution in comparison with some other generalizations of Lomax distribution. It is observed that Lomax-inverse exponential distribution is more robust than the competing distributions, inverse exponential and Lomax distributions. This is an evident that the Lomax generator is a good probability model.

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Zubair, Muhammad, Ayman Alzaatreh, Gauss Cordeiro, M. H. Tahir, and Muhammad Mansoor. "On generalized classes of exponential distribution using T-X family framework." Filomat 32, no. 4 (2018): 1259–72. http://dx.doi.org/10.2298/fil1804259z.

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We introduce new generalized classes of exponential distribution, called T-exponential {Y} class using the quantile functions of well-known distributions. We derive some general mathematical properties of this class including explicit expressions for the quantile function, Shannon entropy, moments and mean deviations. Some generalized exponential families are investigated. The shapes of the models in these families can be symmetric, left-skewed, right-skewed and reversed-J, and the hazard rate can be increasing, decreasing, bathtub, upside-down bathtub, J and reverse-J shaped. Two real data sets are used to illustrate the applicability of the new models.

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Abouelmagd, T. H. M. "The Logarithmic Burr-Hatke Exponential Distribution for Modeling Reliability and Medical Data." International Journal of Statistics and Probability 7, no. 5 (August 9, 2018): 73. http://dx.doi.org/10.5539/ijsp.v7n5p73.

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In this work, we introduced a new one-parameter exponential distribution. Some of its structural properties are derived% \textbf{.} The maximum likelihood method is used to estimate the model parameters by means of numerical Monte Carlo simulation study. The justification for the practicality of the new lifetime model is based on the wider use of the exponential model. The new model can be viewed as a mixtureof the exponentiated exponential distribution. It can also be considered as a suitable model for fitting right skewed data.\textbf{\ }We prove empirically the importance and flexibility of the new model in modelingcancer patients data, the new model provides adequate fits as compared to other related models with small values for $W^{\ast }$\ \ and $A^{\ast }$. The new model is much better than the Modified beta-Weibull, Weibull, exponentiated transmuted generalized Rayleig, the transmuted modified-Weibull, and transmuted additive Weibull models in modeling cancer patients data. We are also motivated to introduce this new model because it has only one parameter and we can generate some new families based on it such as the the odd Burr-Hatke exponential-G family of distributions, the logarithmic\textbf{\ }Burr-Hatke exponential-G family of distributions and the generalized\textbf{\ }Burr-Hatke exponential-G family of distributions, among others.

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Bilal, Muhammad, Muhammad Mohsin, and Muhammad Aslam. "Weibull-Exponential Distribution and Its Application in Monitoring Industrial Process." Mathematical Problems in Engineering 2021 (March 26, 2021): 1–13. http://dx.doi.org/10.1155/2021/6650237.

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This paper presents a new Weibull family of distributions. The compatibility of the newly developed class is justified through its application in the field of quality control using Weibull-exponential distribution, a special case of the proposed family. In this paper, an attribute control chart using Weibull-exponential distribution is developed. The estimations of the model parameters and the proposed chart parameters are performed through the methods of maximum likelihood and average run-length. The significance of the proposed model is demonstrated using a simulation study and real-life problems. The results of the monitoring process and quick detection are compared with exponential distribution.

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Cao, Limei, Huafei Sun, and Xiaojie Wang. "The geometric structures of the Weibull distribution manifold and the generalized exponential distribution manifold." Tamkang Journal of Mathematics 39, no. 1 (March 31, 2008): 45–51. http://dx.doi.org/10.5556/j.tkjm.39.2008.44.

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Investigating the geometric structures of the distribution manifolds is a basic task in information geometry. However, by so far, most works are on the distribution manifolds of exponential family. In this paper, we investigate two non-exponential distribution manifolds —the Weibull distribution manifold and the generalized exponential distribution manifold. Then we obtain their geometric structures.

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Tzougas, George, and Dimitris Karlis. "AN EM ALGORITHM FOR FITTING A NEW CLASS OF MIXED EXPONENTIAL REGRESSION MODELS WITH VARYING DISPERSION." ASTIN Bulletin 50, no. 2 (May 2020): 555–83. http://dx.doi.org/10.1017/asb.2020.13.

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AbstractRegression modelling involving heavy-tailed response distributions, which have heavier tails than the exponential distribution, has become increasingly popular in many insurance settings including non-life insurance. Mixed Exponential models can be considered as a natural choice for the distribution of heavy-tailed claim sizes since their tails are not exponentially bounded. This paper is concerned with introducing a general family of mixed Exponential regression models with varying dispersion which can efficiently capture the tail behaviour of losses. Our main achievement is that we present an Expectation-Maximization (EM)-type algorithm which can facilitate maximum likelihood (ML) estimation for our class of mixed Exponential models which allows for regression specifications for both the mean and dispersion parameters. Finally, a real data application based on motor insurance data is given to illustrate the versatility of the proposed EM-type algorithm.

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Mian, Rajibul, and Sudhir Paul. "Tests of exponentiality against some parametric over/under-dispersed life time models." Acta et Commentationes Universitatis Tartuensis de Mathematica 21, no. 2 (December 22, 2017): 207–23. http://dx.doi.org/10.12697/acutm.2017.21.14.

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We develop tests of goodness of fit of the exponential model against some over/under dispersion family of distributions. In particular, we develop 3 score test statistics and 3 likelihood ratio statistics. These are (S1, L1), (S2, L2), and (S3, L3) based on a general over-dispersed family of distributions, two specic over/under dispersed exponential models, namely, the gamma and the Weibull distributions, respectively. A simulation study shows that the statistics S3 and L3 have best overall performance, in terms of both, level and power. However, the statistic L3 can be liberal in some instances and it needs the maximum likelihood estimates of the parameters of the Weibull distribution as opposed to the statistic S3 which is very simple to use. So, our recommendation is to use the statistic S3 to test the fit of an exponential distribution over any over/under-dispersed exponential distribution.

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David Sam Jayakumar, G. S., A. Solairaju, and A. Sulthan. "A form of multivariate generalised exponential family of distribution." Journal of Statistics and Management Systems 20, no. 5 (September 3, 2017): 847–70. http://dx.doi.org/10.1080/09720510.2017.1401798.

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15

Habibi, Reza. "Change Points with Linear Trend in Exponential Family Distribution." Calcutta Statistical Association Bulletin 55, no. 3-4 (September 2004): 181–98. http://dx.doi.org/10.1177/0008068320040304.

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Jia, Yuheng, Sam Kwong, and Ran Wang. "Applying Exponential Family Distribution to Generalized Extreme Learning Machine." IEEE Transactions on Systems, Man, and Cybernetics: Systems 50, no. 5 (May 2020): 1794–804. http://dx.doi.org/10.1109/tsmc.2017.2788005.

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Baharith, Lamya A., Kholod M. AL-Beladi, and Hadeel S. Klakattawi. "The Odds Exponential-Pareto IV Distribution: Regression Model and Application." Entropy 22, no. 5 (April 25, 2020): 497. http://dx.doi.org/10.3390/e22050497.

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This article introduces the odds exponential-Pareto IV distribution, which belongs to the odds family of distributions. We studied the statistical properties of this new distribution. The odds exponential-Pareto IV distribution provided decreasing, increasing, and upside-down hazard functions. We employed the maximum likelihood method to estimate the distribution parameters. The estimators performance was assessed by conducting simulation studies. A new log location-scale regression model based on the odds exponential-Pareto IV distribution was also introduced. Parameter estimates of the proposed model were obtained using both maximum likelihood and jackknife methods for right-censored data. Real data sets were analyzed under the odds exponential-Pareto IV distribution and log odds exponential-Pareto IV regression model to show their flexibility and potentiality.

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Celikkanat, Abdulkadir, and Fragkiskos D. Malliaros. "Exponential Family Graph Embeddings." Proceedings of the AAAI Conference on Artificial Intelligence 34, no. 04 (April 3, 2020): 3357–64. http://dx.doi.org/10.1609/aaai.v34i04.5737.

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Representing networks in a low dimensional latent space is a crucial task with many interesting applications in graph learning problems, such as link prediction and node classification. A widely applied network representation learning paradigm is based on the combination of random walks for sampling context nodes and the traditional Skip-Gram model to capture center-context node relationships. In this paper, we emphasize on exponential family distributions to capture rich interaction patterns between nodes in random walk sequences. We introduce the generic exponential family graph embedding model, that generalizes random walk-based network representation learning techniques to exponential family conditional distributions. We study three particular instances of this model, analyzing their properties and showing their relationship to existing unsupervised learning models. Our experimental evaluation on real-world datasets demonstrates that the proposed techniques outperform well-known baseline methods in two downstream machine learning tasks.

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McLeish, Don L. "Bounded Relative Error Importance Sampling and Rare Event Simulation." ASTIN Bulletin 40, no. 1 (May 2010): 377–98. http://dx.doi.org/10.2143/ast.40.1.2049235.

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AbstractWe consider estimating tail events using exponential families of importance sampling distributions. When the cannonical sufficient statistic for the exponential family mimics the tail behaviour of the underlying cumulative distribution function, we can achieve bounded relative error for estimating tail probabilities. Examples of rare event simulation from various distributions including Tukey's g&h distribution are provided.

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Rao, R. Prabhakar, and B. C. Sutradhar. "A Global Test for the Goodness of Fit of Generalized Linear Models : An Estimating Equation Approach." Calcutta Statistical Association Bulletin 56, no. 1-4 (March 2005): 251–82. http://dx.doi.org/10.1177/0008068320050514.

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Summary Generalized linear models are used to analyze a wide variety of discrete and continuous data with possible overdispersion under the assumption that the data follow an exponential family of distributions. The violation of this assumption may have adverse effects on the statistical inferences. The existing goodness of fit tests for checking this assumption are valid only for a standard exponential family of distributions with no overdispersion. In this paper, we develop a global goodness of fit test for the general exponential family of distributions which may or may not contain overdispersion. The proposed statistic has asymptotically standard Gaussian distribution which should be easy to implement.

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Al-Babtain, Abdulhakim A., Ibrahim Elbatal, Hazem Al-Mofleh, Ahmed M. Gemeay, Ahmed Z. Afify, and Abdullah M. Sarg. "The Flexible Burr X-G Family: Properties, Inference, and Applications in Engineering Science." Symmetry 13, no. 3 (March 14, 2021): 474. http://dx.doi.org/10.3390/sym13030474.

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In this paper, we introduce a new flexible generator of continuous distributions called the transmuted Burr X-G (TBX-G) family to extend and increase the flexibility of the Burr X generator. The general statistical properties of the TBX-G family are calculated. One special sub-model, TBX-exponential distribution, is studied in detail. We discuss eight estimation approaches to estimating the TBX-exponential parameters, and numerical simulations are conducted to compare the suggested approaches based on partial and overall ranks. Based on our study, the Anderson–Darling estimators are recommended to estimate the TBX-exponential parameters. Using two skewed real data sets from the engineering sciences, we illustrate the importance and flexibility of the TBX-exponential model compared with other existing competing distributions.

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Al-Babtain, Abdulhakim A., Mohammed K. Shakhatreh, Mazen Nassar, and Ahmed Z. Afify. "A New Modified Kies Family: Properties, Estimation Under Complete and Type-II Censored Samples, and Engineering Applications." Mathematics 8, no. 8 (August 12, 2020): 1345. http://dx.doi.org/10.3390/math8081345.

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In this paper, we introduce a new family of continuous distributions that is called the modified Kies family of distributions. The main mathematical properties of the new family are derived. A special case of the new family has been considered in more detail; namely, the two parameters modified Kies exponential distribution with bathtub shape, decreasing and increasing failure rate function. The importance of the new distribution comes from its ability in modeling positively and negatively skewed real data over some generalized distributions with more than two parameters. The shape behavior of the hazard rate and the mean residual life functions of the modified Kies exponential distribution are discussed. We use the method of maximum likelihood to estimate the distribution parameters based on complete and type-II censored samples. The approximate confidence intervals are also obtained under the two schemes. A simulation study is conducted and two real data sets from the engineering field are analyzed to show the flexibility of the new distribution in modeling real life data.

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Reyes, Jimmy, Emilio Gómez-Déniz, Héctor W. Gómez, and Enrique Calderín-Ojeda. "A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory." Symmetry 13, no. 4 (April 14, 2021): 679. http://dx.doi.org/10.3390/sym13040679.

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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 bimodal nature of incorporating covariates in the model in a simple way. Some empirical datasets with positive support, such as losses in insurance portfolios, show an excess of zero values and bimodality. For these cases, classical distributions, such as exponential, gamma, Weibull, or inverse Gaussian, to name a few, are unable to explain data of this nature. This paper attempts to fill this gap in the literature by introducing a family of distributions that can be unimodal or bimodal and nests the exponential distribution. Some of its more relevant properties, including moments, kurtosis, Fisher’s asymmetric coefficient, and several estimation methods, are illustrated. Different results that are related to finance and insurance, such as hazard rate function, limited expected value, and the integrated tail distribution, among other measures, are derived. Because of the simplicity of the mean of this distribution, a regression model is also derived. Finally, examples that are based on actuarial data are used to compare this new family with the exponential distribution.

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Marshall, Albert W., and Ingram Olkin. "Bivariate life distributions from Pólya's urn model for contagion." Journal of Applied Probability 30, no. 3 (September 1993): 497–508. http://dx.doi.org/10.2307/3214760.

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Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

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Marshall, Albert W., and Ingram Olkin. "Bivariate life distributions from Pólya's urn model for contagion." Journal of Applied Probability 30, no. 03 (September 1993): 497–508. http://dx.doi.org/10.1017/s0021900200044259.

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Shock models based on Poisson processes have been used to derive univariate and multivariate exponential distributions. But in many applications, Poisson processes are not realistic models of physical shock processes because they have independent increments; expanded models that allow for possibly dependent increments are of interest. In this paper, univariate and bivariate Pólya urn schemes are used to derive models of shock sources. The life distributions obtained from these models form a large parametric family that includes the exponential distribution. Even in the univariate case these life distributions have not been widely used, though they form a large and flexible family. In the bivariate case, the family includes the bivariate exponential distributions of Marshall and Olkin as a special case.

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del Castillo, Joan. "The singly truncated normal distribution: A non-steep exponential family." Annals of the Institute of Statistical Mathematics 46, no. 1 (March 1994): 57–66. http://dx.doi.org/10.1007/bf00773592.

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Chesneau, Christophe, Hassan S. Bakouch, and Muhammad Nauman Khan. "A weighted transmuted exponential distributions with environmental applications." Statistics, Optimization & Information Computing 8, no. 1 (February 17, 2020): 36–53. http://dx.doi.org/10.19139/soic-2310-5070-785.

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In this paper, we introduce a new three-parameter distribution. It is based on the combination of a re-parametrization of a general family of distributions (known as the EGNB2 distribution) and the so-called quadratic rank transmutation map defined with the exponential distribution as baseline. We explore some mathematical properties of this distribution including the hazard rate function, moments, the moment generating function, the quantile function, various entropy measures and (reversed) residual life functions. A statistical study investigates estimation of the parameters using the method of maximum likelihood. The distribution along with other existing distributions are fitted to two environmental data sets and its superior performance is assessed by using some goodness-of-fit tests. As a result, some environmental measures associated with these data are obtained such as the return level and mean deviation about this level.

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García, Victoriano, María Martel-Escobar, and F. J. Vázquez-Polo. "Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure." Symmetry 12, no. 3 (March 15, 2020): 464. http://dx.doi.org/10.3390/sym12030464.

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This paper presents a three-parameter family of distributions which includes the common exponential and the Marshall–Olkin exponential as special cases. This distribution exhibits a monotone failure rate function, which makes it appealing for practitioners interested in reliability, and means it can be included in the catalogue of appropriate non-symmetric distributions to model these issues, such as the gamma and Weibull three-parameter families. Given the lack of symmetry of this kind of distribution, various statistical and reliability properties of this model are examined. Numerical examples based on real data reflect the suitable behaviour of this distribution for modelling purposes.

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Al-Marzouki, Sanaa, and Sharifah Alrajhi. "A New-Flexible Generated Family of Distributions Based on Half-Logistic Distribution." Journal of Computational and Theoretical Nanoscience 17, no. 11 (November 1, 2020): 4813–18. http://dx.doi.org/10.1166/jctn.2020.9332.

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We proposed a new family of distributions from a half logistic model called the generalized odd half logistic family. We expressed its density function as a linear combination of exponentiated densities. We calculate some statistical properties as the moments, probability weighted moment, quantile and order statistics. Two new special models are mentioned. We study the estimation of the parameters for the odd generalized half logistic exponential and the odd generalized half logistic Rayleigh models by using maximum likelihood method. One real data set is assesed to illustrate the usefulness of the subject family.

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Hemeda, Saeed E., and Ali M. Abdallah. "Sinh Inverted Exponential Distribution: Simulation & Application to Neck Cancer Disease." International Journal of Statistics and Probability 9, no. 5 (July 27, 2020): 11. http://dx.doi.org/10.5539/ijsp.v9n5p11.

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A goal of this research is providing new probability distribution called Sinh inverted exponential distribution. The new distribution was extensively depending on the hyperbolic sine family of distributions with exponential distribution as a baseline distribution. Valuable statistical properties of the proposed distribution including mathematical and asymptotic expressions for its probability density function and Reliability. Moments, quantiles, moment generating function, failure rate function, mean residual lifetime, order statistics and entropies are derived. Actually, the applicability and validation of this model is proved in simulation study and an application to neck cancer disease data.

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Nadarajah, Saralees, and Samuel Kotz. "Reliability for some bivariate exponential distributions." Mathematical Problems in Engineering 2006 (2006): 1–14. http://dx.doi.org/10.1155/mpe/2006/41652.

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In the area of stress-strength models, there has been a large amount of work as regards estimation of the reliabilityR=Pr⁡(X<Y). The algebraic form forR=Pr⁡(X<Y)has been worked out for the vast majority of the well-known distributions whenXandYare independent random variables belonging to the same univariate family. In this paper, forms ofRare considered when(X,Y)follow bivariate distributions with dependence betweenXandY. In particular, explicit expressions forRare derived when the joint distribution isbivariate exponential. The calculations involve the use of special functions. An application of the results is also provided.

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Bantan, Rashad A. R., Christophe Chesneau, Farrukh Jamal, Ibrahim Elbatal, and Mohammed Elgarhy. "The Truncated Burr X-G Family of Distributions: Properties and Applications to Actuarial and Financial Data." Entropy 23, no. 8 (August 21, 2021): 1088. http://dx.doi.org/10.3390/e23081088.

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In this article, the “truncated-composed” scheme was applied to the Burr X distribution to motivate a new family of univariate continuous-type distributions, called the truncated Burr X generated family. It is mathematically simple and provides more modeling freedom for any parental distribution. Additional functionality is conferred on the probability density and hazard rate functions, improving their peak, asymmetry, tail, and flatness levels. These characteristics are represented analytically and graphically with three special distributions of the family derived from the exponential, Rayleigh, and Lindley distributions. Subsequently, we conducted asymptotic, first-order stochastic dominance, series expansion, Tsallis entropy, and moment studies. Useful risk measures were also investigated. The remainder of the study was devoted to the statistical use of the associated models. In particular, we developed an adapted maximum likelihood methodology aiming to efficiently estimate the model parameters. The special distribution extending the exponential distribution was applied as a statistical model to fit two sets of actuarial and financial data. It performed better than a wide variety of selected competing non-nested models. Numerical applications for risk measures are also given.

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Mirzadeh, Saeed, and Anis Iranmanesh. "A new class of skew-logistic distribution." Mathematical Sciences 13, no. 4 (October 5, 2019): 375–85. http://dx.doi.org/10.1007/s40096-019-00306-8.

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Abstract In this study, the researchers introduce a new class of the logistic distribution which can be used to model the unimodal data with some skewness present. The new generalization is carried out using the basic idea of Nadarajah (Statistics 48(4):872–895, 2014), called truncated-exponential skew-logistic (TESL) distribution. The TESL distribution is a member of the exponential family; therefore, the skewness parameter can be derived easier. Meanwhile, some important statistical characteristics are presented; the real data set and simulation studies are applied to evaluate the results. Also, the TESL distribution is compared to at least five other skew-logistic distributions.

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Sapkota, Laxmi Prasad. "Exponentiated–Exponential Logistic Distribution: Some Properties and Application." Janapriya Journal of Interdisciplinary Studies 9, no. 1 (December 31, 2020): 100–108. http://dx.doi.org/10.3126/jjis.v9i1.35280.

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This study proposes new distribution which is generated from exponentiated-exponential-X family of distribution. It is explored various shape and behavior of the observed distribution through probability density plot, hazard rate function and quantile function. Further we have investigated some mathematical properties, estimation of the parameters and associated confidence interval using maximum likelihood estimation (MLE) method of the exponentiatedexponential-logistic distribution (EELD).

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Siegrist, Kyle. "RANDOM FINITE SUBSETS WITH EXPONENTIAL DISTRIBUTIONS." Probability in the Engineering and Informational Sciences 21, no. 1 (December 15, 2006): 117–32. http://dx.doi.org/10.1017/s0269964807070088.

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Let S denote the collection of all finite subsets of . We define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural home for probability distributions with exponential properties, such as the memoryless and constant rate properties. We show that there are no exponential distributions on S, but that S can be partitioned into subsemigroups, each of which supports a one-parameter family of exponential distributions. We then find the distribution on S that is closest to exponential, in a certain sense. This work might have applications to the problem of selecting a finite sample from a countably infinite population in the most random way.

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Huo, Xiaoyan, Saima K. Khosa, Zubair Ahmad, Zahra Almaspoor, Muhammad Ilyas, and Muhammad Aamir. "A New Lifetime Exponential-X Family of Distributions with Applications to Reliability Data." Mathematical Problems in Engineering 2020 (August 18, 2020): 1–16. http://dx.doi.org/10.1155/2020/1316345.

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Modeling reliability data with nonmonotone hazards is a prominent research topic that is quite rich and still growing rapidly. Many studies have suggested introducing new families of distributions to modify the Weibull distribution to model the nonmonotone hazards. In the present study, we propose a new family of distributions called a new lifetime exponential-X family. A special submodel of the proposed family called a new lifetime exponential-Weibull distribution suitable for modeling reliability data with bathtub-shaped hazard rates is discussed. The maximum-likelihood estimators of the model parameters are obtained. A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators. For illustrative purposes, two real applications from reliability engineering with bathtub-shaped hazard functions are analyzed. The practical applications show that the proposed model provides better fits than the other nonnested models.

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Al-Marzouki, Sanaa, and Sharifah Alrajhi. "The Odd Generalized NH Inverse Exponential Model: Theory and Application." Journal of Computational and Theoretical Nanoscience 17, no. 11 (November 1, 2020): 4835–40. http://dx.doi.org/10.1166/jctn.2020.9407.

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In this paper we introduce a new lifetime distribution derived from odd generalized NH-G (OGNH-G) family technique called OGNH-inverse exponential (OGNHIE) distribution. We establish various mathematical properties. The maximum likelihood (ML) estimates for the OGNHIE parameters are derived. Finally the model is applied to a real dataset. We apply goodness of fit statistics and graphical tools to examine the adequacy of the OGNHIE distribution. The importance of this research lies in deriving a new distribution under the name OGNHIE, which is considered the best distributions in analyzing data of life times at present if compared to many distribution in analysis real data.

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Vere-Jones, David. "A limit theorem with application to Båth's law in seismology." Advances in Applied Probability 40, no. 03 (September 2008): 882–96. http://dx.doi.org/10.1017/s0001867800002834.

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In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

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Vere-Jones, David. "A limit theorem with application to Båth's law in seismology." Advances in Applied Probability 40, no. 3 (September 2008): 882–96. http://dx.doi.org/10.1239/aap/1222868190.

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In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

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40

Iqbal, Zafar, Muhammad Wasim, and Naureen Riaz. "EXPONENTIATED MOMENT EXPONENTIAL DISTRIBUTION AND POWER SERIES DISTRIBUTION WITH APPLICATIONS: A NEW COMPOUND FAMILY." International Journal of Advanced Research 5, no. 7 (July 31, 2017): 1335–55. http://dx.doi.org/10.21474/ijar01/4844.

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Bantan, Rashad A. R., Farrukh Jamal, Christophe Chesneau, and Mohammed Elgarhy. "A New Power Topp–Leone Generated Family of Distributions with Applications." Entropy 21, no. 12 (November 29, 2019): 1177. http://dx.doi.org/10.3390/e21121177.

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In this paper, we introduce a new general family of distributions obtained by a subtle combination of two well-established families of distributions: the so-called power Topp–Leone-G and inverse exponential-G families. Its definition is centered around an original cumulative distribution function involving exponential and polynomial functions. Some desirable theoretical properties of the new family are discussed in full generality, with comprehensive results on stochastic ordering, quantile function and related measures, general moments and related measures, and the Shannon entropy. Then, a statistical parametric model is constructed from a special member of the family, defined with the use of the inverse Lomax distribution as the baseline distribution. The maximum likelihood method was applied to estimate the unknown model parameters. From the general theory of this method, the asymptotic confidence intervals of these parameters were deduced. A simulation study was conducted to evaluate the numerical behavior of the estimates we obtained. Finally, in order to highlight the practical perspectives of the new family, two real-life data sets were analyzed. All the measures considered are favorable to the new model in comparison to four serious competitors.

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Ma, Di, and Songcan Chen. "Distribution agnostic Bayesian matching pursuit based on the exponential embedded family." Neurocomputing 410 (October 2020): 401–9. http://dx.doi.org/10.1016/j.neucom.2020.06.007.

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Jayakumar, G. S. David Sam, A. Solairaju, and A. Sulthan. "A Multivariate Generalized Double Exponential Family of Distribution of Kind-1." Journal of Statistics and Management Systems 17, no. 5-6 (November 2, 2014): 445–78. http://dx.doi.org/10.1080/09720510.2013.876772.

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COX, D. R. "A note on design when response has an exponential family distribution." Biometrika 75, no. 1 (1988): 161–64. http://dx.doi.org/10.1093/biomet/75.1.161.

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Okasha, Hassan M., and M. Kayid. "A new family of Marshall–Olkin extended generalized linear exponential distribution." Journal of Computational and Applied Mathematics 296 (April 2016): 576–92. http://dx.doi.org/10.1016/j.cam.2015.10.017.

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Yamada, Hiroaki S., and Kazumoto Iguchi. "-exponential fitting for distributions of family names." Physica A: Statistical Mechanics and its Applications 387, no. 7 (March 2008): 1628–36. http://dx.doi.org/10.1016/j.physa.2007.11.002.

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Letac, Gérard, and V. Seshadri. "Haight's distributions as a natural exponential family." Statistics & Probability Letters 6, no. 3 (February 1988): 165–69. http://dx.doi.org/10.1016/0167-7152(88)90115-0.

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48

Lee, Jaehyung, Mark S. Kaiser, and Noel Cressie. "Multiway Dependence in Exponential Family Conditional Distributions." Journal of Multivariate Analysis 79, no. 2 (November 2001): 171–90. http://dx.doi.org/10.1006/jmva.2000.1966.

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Zhang, Fode, Xiaolin Shi, and Hon Keung Tony Ng. "Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring." Entropy 23, no. 6 (May 28, 2021): 687. http://dx.doi.org/10.3390/e23060687.

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In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

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Bantan, Rashad A. R., Christophe Chesneau, Farrukh Jamal, and Mohammed Elgarhy. "On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions." Mathematics 8, no. 6 (June 11, 2020): 953. http://dx.doi.org/10.3390/math8060953.

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This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes. It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function. Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard. Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics. Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines. After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the “unexponentiated” version of the proposed model as well. As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.

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