INDRANIL GHOSH - Academia.edu (original) (raw)
Papers by INDRANIL GHOSH
Pakistan Journal of Statistics and Operation Research
We would like to point out that the formula for the cumulative distribution given in Coelho-Barro... more We would like to point out that the formula for the cumulative distribution given in Coelho-Barros et al. (2017) and a similar version of it given in Usman et al. (2021) are not cumulative distribution functions as these functions do not satisfy the one or more necessary and sufficient conditions for a function to be a cumulative distribution function. We would also like to mention that formulas for the cumulative distribution functions of product and ratio of two independent Pareto and Exponential random variables given by Obeid and Kadry (2022) are not cumulative distribution functions either. We do not believe that these formulas can be fixed to be cumulative distribution functions. In this short article, we provide mathematical justification in support of these claims.
Journal of Risk and Financial Management
Copulas are a quite flexible and useful tool for modeling the dependence structure between two or... more Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate.
Mathematics
A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is d... more A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is developed and discussed as a novel probability model. We study some useful structural properties of the proposed model. To estimate the model parameters under the classical method, we use the maximum likelihood estimation using a progressive type II censoring scheme. Under the Bayesian paradigm the estimation is carried out with gamma priors under a progressive type II censored samples with squared error loss function. To demonstrate the efficiency of the proposed model based on progressively type II censoring, a simulation study is carried out. Three actual data sets are used as an example, demonstrating that the suggested model in the new class fits better than the existing finite mixture models available in the literature.
• In this paper we explore some mechanisms for constructing bivariate and multivariate beta and K... more • In this paper we explore some mechanisms for constructing bivariate and multivariate beta and Kumaraswamy distributions. Specifically, we focus our attention on the Arnold-Ng (2011) eight parameter bivariate beta model. Several models in the literature are identified as special cases of this distribution including the Jones-Olkin-Liu-Libby-Novick bivariate beta model, and certain Kotz and Nadarajah bivariate beta models among others. The utility of such models in constructing bivariate Kumaraswamy models is investigated. Structural properties of such derived models are studied. Parameter estimation for the models is also discussed. For illustrative purposes, a real life data set is considered to exhibit the applicability of these models in comparison with rival bivariate beta and Kumaraswamy models.
International Journal of Statistics and Probability, 2015
In this paper, we establish certain characterizations of the Weibull-X family of distributions pr... more In this paper, we establish certain characterizations of the Weibull-X family of distributions proposed by Alzaatreh et al. (2013) as well as of the Burr XII Negative Binomial distribution, introduced by Ramos et al. (2015). These characterizations are based on two truncated moments, hazard rate function and conditional expectation of functions of random variables.
Mathematics
A new family of distributions called the mixture of the exponentiated Kumaraswamy-G (henceforth, ... more A new family of distributions called the mixture of the exponentiated Kumaraswamy-G (henceforth, in short, ExpKum-G) class is developed. We consider Weibull distribution as the baseline (G) distribution to propose and study this special sub-model, which we call the exponentiated Kumaraswamy Weibull distribution. Several useful statistical properties of the proposed ExpKum-G distribution are derived. Under the classical paradigm, we consider the maximum likelihood estimation under progressive type II censoring to estimate the model parameters. Under the Bayesian paradigm, independent gamma priors are proposed to estimate the model parameters under progressive type II censored samples, assuming several loss functions. A simulation study is carried out to illustrate the efficiency of the proposed estimation strategies under both classical and Bayesian paradigms, based on progressively type II censoring models. For illustrative purposes, a real data set is considered that exhibits that ...
Journal of Statistical Theory and Applications, 2018
are derived when X 1 and X 2 are independent or sub-independent Kumaraswamy random variables. The... more are derived when X 1 and X 2 are independent or sub-independent Kumaraswamy random variables. The expressions appear to involve the incomplete gamma functions. Some possible real life scenarios are mentioned in which such quantities might be of interest.
Pakistan Journal of Statistics and Operation Research, 2020
We study a new family of distributions defined by the minimum of the Poissonrandom number of inde... more We study a new family of distributions defined by the minimum of the Poissonrandom number of independent identically distributed random variables having a general Weibull-G distribution (see Bourguignon et al. (2014)). Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics, reliability and entropies are derived. Maximum likelihood estimation of the model parameters is investigated. Three special models of the new family are discussed. We perform three applications to real data sets to show the potentiality of theproposed family.
Austrian Journal of Statistics, 2017
We define and study a new generalization of the Fréchet distribution called the beta exponential ... more We define and study a new generalization of the Fréchet distribution called the beta exponential Fréchet distribution. The new model includes thirty two special models. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean residual life, mean inactivity time, order statistics and entropies are derived. The method of maximum likelihood is proposed to estimate the model parameters. A small simulation study is also<br />reported. Two real data sets are applied to illustrate the flexibility of the proposed model compared with some nested and non-nested models.
Conditional specification of distributions is a developing area with increasing applications. In ... more Conditional specification of distributions is a developing area with increasing applications. In the finite discrete case, a variety of compatible conditions can be derived. In this paper, we propose an alternative approach to study the compatibility of two conditional probability distributions under the finite discrete setup. A technique based on rank-based criterion is shown to be particularly convenient for identifying compatible distributions corresponding to complete conditional specification including the case with zeros.The proposed methods are illustrated with several examples.
Mathematics, 2021
Recently, there seems to be an increasing amount of interest in the use of the tail conditional e... more Recently, there seems to be an increasing amount of interest in the use of the tail conditional expectation (TCE) as a useful measure of risk associated with a production process, for example, in the measurement of risk associated with stock returns corresponding to the manufacturing industry, such as the production of electric bulbs, investment in housing development, and financial institutions offering loans to small-scale industries. Companies typically face three types of risk (and associated losses from each of these sources): strategic (S); operational (O); and financial (F) (insurance companies additionally face insurance risks) and they come from multiple sources. For asymmetric and bounded losses (properly adjusted as necessary) that are continuous in nature, we conjecture that risk assessment measures via univariate/bivariate Kumaraswamy distribution will be efficient in the sense that the resulting TCE based on bivariate Kumaraswamy type copulas do not depend on the margi...
Communications in Statistics - Theory and Methods
ABSTRACT It is also shown that our proposed skew-normal model subsumes many other well-known skew... more ABSTRACT It is also shown that our proposed skew-normal model subsumes many other well-known skew-normal model that exists in the literature. Recent work on a new two-parameter generalized skew-normal model has received a lot of attention. This paper presents a new generalized Balakrishnan type skew–normal distribution by introducing two shape parameters. We also provide some useful results for this new generalization. It is also shown that our proposed skew–normal model subsumes the original Balakrishnan skew–normal model (2002) as well as other well–known skew–normal models as special cases. The resulting flexible model can be expected to fit a wider variety of data structures than either of the models involving a single skewing mechanism. For illustrative purposes, a famed data set on IQ scores has been used to exhibit the efficacy of the proposed model.
We propose and study a new class of continuous distributions called the beta WeibullG family whic... more We propose and study a new class of continuous distributions called the beta WeibullG family which extends the Weibull-G family introduced by Bourguignon et al. (2014). Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, order statistics and probability weighted moments are derived. The maximum likelihood is used for estimating the model parameters. The importance and flexibility of the new family are illustrated by means of two applications to real data sets.
Journal of Data Science
In the literature, several continuous univariate distributions have been extensively used for mod... more In the literature, several continuous univariate distributions have been extensively used for modeling data in many areas such as economics, engineering, biological studies and environmental sciences. However, applied areas such as …nance, lifetime analysis and insurance clearly require extended forms of these distributions. So, several classes of distributions have been constructed by extending common families of continuous distributions. These generalized distributions give more ‡exibility by adding one "or more" parameters to the baseline model. They were pioneered by Gupta et al. (1998) who proposed the exponentiated-G class, which consists of raising the cumulative distribution function (cdf) to a positive power parameter. Many other classes can be cited such as the Marshall-Olkin-G family by Marshall and Olkin (1997), beta generalized-G family by Eugene et al. (2002), exponentiated generalized-G family by Cordeiro et al. (2013), a new method for generating families of continuous distributions by Alzaatreh et al. (2013), transmuted exponentiated generalized-G by Yousof et al. (2015), exponentiated transmuted-G by Merovc et al. (2016), Burr X-G by Yousof et al. (2016), transmuted Weibull G family by Alizadeh et al. (2016), complementary generalized transmuted Poisson-G family by Alizadeh et al. (2016b), transmuted geometric-G by A…fy et al. (2016a), complementary geometric transmuted-G family A…fy et al. (2016b), Kumaraswamy transmuted-G by A…fy et al. (2016c), exponentiated generalized-G Poisson by Aryal and Yousof (2017), Marshall-Olkin generalized family by Yousof et al. (2017a), beta Weibull-G family of distributions by Yousof et al. (2017b), Type I general exponential class of distributions by Hamedani et al. (2017), Topp-Leone odd log-logistic family by de Brito et al. (2017), generalized odd generalized exponential family by Alizadeh et al. (2017), exponentiated Weibull-H family Cordeiro et al. (2017a), generalized transmuted-G by Nofal et al. (2017), Burr XII system of densities by Cordeiro et al. (2017b) and beta transmuted-H family by A…fy et al. (2017), among others.
American Journal of Mathematical and Management Sciences
Abstract In this article, we try to supplement the distribution theory literature by incorporatin... more Abstract In this article, we try to supplement the distribution theory literature by incorporating a new bounded distribution, called the bounded weighted exponential (BWE) distribution in the (0, 1) intervals by transformation method. The proposed distribution exhibits decreasing and left-skewed unimodal density while the hazard rate can have increasing and bathtub shaped. Although our main focus is on the estimation from the frequentist point of view, in addition, we derive some useful structural and statistical properties of the proposed BWE distribution. We briefly describe three classical estimators namely, the maximum likelihood estimators (MLE), the ordinary least-squares estimators (LSE) and the weighted least-squares estimators (WLSE). Monte Carlo simulations are performed to compare performances of the proposed methods of estimation for both moderate and large samples. An application of the model is presented by re-analyzing a real data set and comparisons are made with the fit attained by some other well-known distributions for illustrative purposes.
In this paper, the Weibull-X family is proposed and some of its properties are discussed. A membe... more In this paper, the Weibull-X family is proposed and some of its properties are discussed. A member of the Weibull-X family, the Weibull-logistic distribution, is defined and studied. Various properties of the Weibull-logistic distribution are obtained. The distribution is found to be unimodal and the shape can be symmetric, right skewed or left skewed. The structural analysis of the distribution in this paper includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy and order statistics. The method of maximum likelihood estimation is proposed for estimating the model parameters. A real data set is used to illustrate the application of the Weibull-logistic distribution.
arXiv: Statistics Theory, 2018
A two parameter discrete gamma-Lomax distribution is derived as a discrete analogous to the conti... more A two parameter discrete gamma-Lomax distribution is derived as a discrete analogous to the continuous three parameters gamma-Lomax distribution (see Alzaatreh et al. (2013, 2014)) using the general approach for discretization of continuous probability distributions. Some useful structural properties of the proposed distribution are examined. Possible areas of application are also discussed.
Journal of Data Science
We introduce a new class of distributions called the generalized odd generalized exponential fami... more We introduce a new class of distributions called the generalized odd generalized exponential family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Ŕnyi, Shannon and q-entropies, order statistics and probability weighted moments are derived. We also propose bivariate generalizations. We constructed a simple type Copula and introduced a useful stochastic property. The maximum likelihood method is used for estimating the model parameters. The importance and flexibility of the new family are illustrated by means of two applications to real data sets. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors via a simulation study.
Pakistan Journal of Statistics and Operation Research
We would like to point out that the formula for the cumulative distribution given in Coelho-Barro... more We would like to point out that the formula for the cumulative distribution given in Coelho-Barros et al. (2017) and a similar version of it given in Usman et al. (2021) are not cumulative distribution functions as these functions do not satisfy the one or more necessary and sufficient conditions for a function to be a cumulative distribution function. We would also like to mention that formulas for the cumulative distribution functions of product and ratio of two independent Pareto and Exponential random variables given by Obeid and Kadry (2022) are not cumulative distribution functions either. We do not believe that these formulas can be fixed to be cumulative distribution functions. In this short article, we provide mathematical justification in support of these claims.
Journal of Risk and Financial Management
Copulas are a quite flexible and useful tool for modeling the dependence structure between two or... more Copulas are a quite flexible and useful tool for modeling the dependence structure between two or more variables or components of bivariate and multivariate vectors, in particular, to predict losses in insurance and finance. In this article, we use the VineCopula package in R to study the dependence structure of some well-known real-life insurance data and identify the best bivariate copula in each case. Associated structural properties of these bivariate copulas are also discussed with a major focus on their tail dependence structure. This study shows that certain types of Archimedean copula with the heavy tail dependence property are a reasonable framework to start in terms modeling insurance claim data both in the bivariate as well as in the case of multivariate domains as appropriate.
Mathematics
A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is d... more A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is developed and discussed as a novel probability model. We study some useful structural properties of the proposed model. To estimate the model parameters under the classical method, we use the maximum likelihood estimation using a progressive type II censoring scheme. Under the Bayesian paradigm the estimation is carried out with gamma priors under a progressive type II censored samples with squared error loss function. To demonstrate the efficiency of the proposed model based on progressively type II censoring, a simulation study is carried out. Three actual data sets are used as an example, demonstrating that the suggested model in the new class fits better than the existing finite mixture models available in the literature.
• In this paper we explore some mechanisms for constructing bivariate and multivariate beta and K... more • In this paper we explore some mechanisms for constructing bivariate and multivariate beta and Kumaraswamy distributions. Specifically, we focus our attention on the Arnold-Ng (2011) eight parameter bivariate beta model. Several models in the literature are identified as special cases of this distribution including the Jones-Olkin-Liu-Libby-Novick bivariate beta model, and certain Kotz and Nadarajah bivariate beta models among others. The utility of such models in constructing bivariate Kumaraswamy models is investigated. Structural properties of such derived models are studied. Parameter estimation for the models is also discussed. For illustrative purposes, a real life data set is considered to exhibit the applicability of these models in comparison with rival bivariate beta and Kumaraswamy models.
International Journal of Statistics and Probability, 2015
In this paper, we establish certain characterizations of the Weibull-X family of distributions pr... more In this paper, we establish certain characterizations of the Weibull-X family of distributions proposed by Alzaatreh et al. (2013) as well as of the Burr XII Negative Binomial distribution, introduced by Ramos et al. (2015). These characterizations are based on two truncated moments, hazard rate function and conditional expectation of functions of random variables.
Mathematics
A new family of distributions called the mixture of the exponentiated Kumaraswamy-G (henceforth, ... more A new family of distributions called the mixture of the exponentiated Kumaraswamy-G (henceforth, in short, ExpKum-G) class is developed. We consider Weibull distribution as the baseline (G) distribution to propose and study this special sub-model, which we call the exponentiated Kumaraswamy Weibull distribution. Several useful statistical properties of the proposed ExpKum-G distribution are derived. Under the classical paradigm, we consider the maximum likelihood estimation under progressive type II censoring to estimate the model parameters. Under the Bayesian paradigm, independent gamma priors are proposed to estimate the model parameters under progressive type II censored samples, assuming several loss functions. A simulation study is carried out to illustrate the efficiency of the proposed estimation strategies under both classical and Bayesian paradigms, based on progressively type II censoring models. For illustrative purposes, a real data set is considered that exhibits that ...
Journal of Statistical Theory and Applications, 2018
are derived when X 1 and X 2 are independent or sub-independent Kumaraswamy random variables. The... more are derived when X 1 and X 2 are independent or sub-independent Kumaraswamy random variables. The expressions appear to involve the incomplete gamma functions. Some possible real life scenarios are mentioned in which such quantities might be of interest.
Pakistan Journal of Statistics and Operation Research, 2020
We study a new family of distributions defined by the minimum of the Poissonrandom number of inde... more We study a new family of distributions defined by the minimum of the Poissonrandom number of independent identically distributed random variables having a general Weibull-G distribution (see Bourguignon et al. (2014)). Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics, reliability and entropies are derived. Maximum likelihood estimation of the model parameters is investigated. Three special models of the new family are discussed. We perform three applications to real data sets to show the potentiality of theproposed family.
Austrian Journal of Statistics, 2017
We define and study a new generalization of the Fréchet distribution called the beta exponential ... more We define and study a new generalization of the Fréchet distribution called the beta exponential Fréchet distribution. The new model includes thirty two special models. Some of its mathematical properties, including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean residual life, mean inactivity time, order statistics and entropies are derived. The method of maximum likelihood is proposed to estimate the model parameters. A small simulation study is also<br />reported. Two real data sets are applied to illustrate the flexibility of the proposed model compared with some nested and non-nested models.
Conditional specification of distributions is a developing area with increasing applications. In ... more Conditional specification of distributions is a developing area with increasing applications. In the finite discrete case, a variety of compatible conditions can be derived. In this paper, we propose an alternative approach to study the compatibility of two conditional probability distributions under the finite discrete setup. A technique based on rank-based criterion is shown to be particularly convenient for identifying compatible distributions corresponding to complete conditional specification including the case with zeros.The proposed methods are illustrated with several examples.
Mathematics, 2021
Recently, there seems to be an increasing amount of interest in the use of the tail conditional e... more Recently, there seems to be an increasing amount of interest in the use of the tail conditional expectation (TCE) as a useful measure of risk associated with a production process, for example, in the measurement of risk associated with stock returns corresponding to the manufacturing industry, such as the production of electric bulbs, investment in housing development, and financial institutions offering loans to small-scale industries. Companies typically face three types of risk (and associated losses from each of these sources): strategic (S); operational (O); and financial (F) (insurance companies additionally face insurance risks) and they come from multiple sources. For asymmetric and bounded losses (properly adjusted as necessary) that are continuous in nature, we conjecture that risk assessment measures via univariate/bivariate Kumaraswamy distribution will be efficient in the sense that the resulting TCE based on bivariate Kumaraswamy type copulas do not depend on the margi...
Communications in Statistics - Theory and Methods
ABSTRACT It is also shown that our proposed skew-normal model subsumes many other well-known skew... more ABSTRACT It is also shown that our proposed skew-normal model subsumes many other well-known skew-normal model that exists in the literature. Recent work on a new two-parameter generalized skew-normal model has received a lot of attention. This paper presents a new generalized Balakrishnan type skew–normal distribution by introducing two shape parameters. We also provide some useful results for this new generalization. It is also shown that our proposed skew–normal model subsumes the original Balakrishnan skew–normal model (2002) as well as other well–known skew–normal models as special cases. The resulting flexible model can be expected to fit a wider variety of data structures than either of the models involving a single skewing mechanism. For illustrative purposes, a famed data set on IQ scores has been used to exhibit the efficacy of the proposed model.
We propose and study a new class of continuous distributions called the beta WeibullG family whic... more We propose and study a new class of continuous distributions called the beta WeibullG family which extends the Weibull-G family introduced by Bourguignon et al. (2014). Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, order statistics and probability weighted moments are derived. The maximum likelihood is used for estimating the model parameters. The importance and flexibility of the new family are illustrated by means of two applications to real data sets.
Journal of Data Science
In the literature, several continuous univariate distributions have been extensively used for mod... more In the literature, several continuous univariate distributions have been extensively used for modeling data in many areas such as economics, engineering, biological studies and environmental sciences. However, applied areas such as …nance, lifetime analysis and insurance clearly require extended forms of these distributions. So, several classes of distributions have been constructed by extending common families of continuous distributions. These generalized distributions give more ‡exibility by adding one "or more" parameters to the baseline model. They were pioneered by Gupta et al. (1998) who proposed the exponentiated-G class, which consists of raising the cumulative distribution function (cdf) to a positive power parameter. Many other classes can be cited such as the Marshall-Olkin-G family by Marshall and Olkin (1997), beta generalized-G family by Eugene et al. (2002), exponentiated generalized-G family by Cordeiro et al. (2013), a new method for generating families of continuous distributions by Alzaatreh et al. (2013), transmuted exponentiated generalized-G by Yousof et al. (2015), exponentiated transmuted-G by Merovc et al. (2016), Burr X-G by Yousof et al. (2016), transmuted Weibull G family by Alizadeh et al. (2016), complementary generalized transmuted Poisson-G family by Alizadeh et al. (2016b), transmuted geometric-G by A…fy et al. (2016a), complementary geometric transmuted-G family A…fy et al. (2016b), Kumaraswamy transmuted-G by A…fy et al. (2016c), exponentiated generalized-G Poisson by Aryal and Yousof (2017), Marshall-Olkin generalized family by Yousof et al. (2017a), beta Weibull-G family of distributions by Yousof et al. (2017b), Type I general exponential class of distributions by Hamedani et al. (2017), Topp-Leone odd log-logistic family by de Brito et al. (2017), generalized odd generalized exponential family by Alizadeh et al. (2017), exponentiated Weibull-H family Cordeiro et al. (2017a), generalized transmuted-G by Nofal et al. (2017), Burr XII system of densities by Cordeiro et al. (2017b) and beta transmuted-H family by A…fy et al. (2017), among others.
American Journal of Mathematical and Management Sciences
Abstract In this article, we try to supplement the distribution theory literature by incorporatin... more Abstract In this article, we try to supplement the distribution theory literature by incorporating a new bounded distribution, called the bounded weighted exponential (BWE) distribution in the (0, 1) intervals by transformation method. The proposed distribution exhibits decreasing and left-skewed unimodal density while the hazard rate can have increasing and bathtub shaped. Although our main focus is on the estimation from the frequentist point of view, in addition, we derive some useful structural and statistical properties of the proposed BWE distribution. We briefly describe three classical estimators namely, the maximum likelihood estimators (MLE), the ordinary least-squares estimators (LSE) and the weighted least-squares estimators (WLSE). Monte Carlo simulations are performed to compare performances of the proposed methods of estimation for both moderate and large samples. An application of the model is presented by re-analyzing a real data set and comparisons are made with the fit attained by some other well-known distributions for illustrative purposes.
In this paper, the Weibull-X family is proposed and some of its properties are discussed. A membe... more In this paper, the Weibull-X family is proposed and some of its properties are discussed. A member of the Weibull-X family, the Weibull-logistic distribution, is defined and studied. Various properties of the Weibull-logistic distribution are obtained. The distribution is found to be unimodal and the shape can be symmetric, right skewed or left skewed. The structural analysis of the distribution in this paper includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy and order statistics. The method of maximum likelihood estimation is proposed for estimating the model parameters. A real data set is used to illustrate the application of the Weibull-logistic distribution.
arXiv: Statistics Theory, 2018
A two parameter discrete gamma-Lomax distribution is derived as a discrete analogous to the conti... more A two parameter discrete gamma-Lomax distribution is derived as a discrete analogous to the continuous three parameters gamma-Lomax distribution (see Alzaatreh et al. (2013, 2014)) using the general approach for discretization of continuous probability distributions. Some useful structural properties of the proposed distribution are examined. Possible areas of application are also discussed.
Journal of Data Science
We introduce a new class of distributions called the generalized odd generalized exponential fami... more We introduce a new class of distributions called the generalized odd generalized exponential family. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Ŕnyi, Shannon and q-entropies, order statistics and probability weighted moments are derived. We also propose bivariate generalizations. We constructed a simple type Copula and introduced a useful stochastic property. The maximum likelihood method is used for estimating the model parameters. The importance and flexibility of the new family are illustrated by means of two applications to real data sets. We assess the performance of the maximum likelihood estimators in terms of biases and mean squared errors via a simulation study.