Family Of Continuous Distributions Research Articles

The New Strange Generalized Rayleigh Family: Characteristics and Applications to COVID-19 Data

In this paper, we introduce a novel family of continuous distributions known as the Odd Generalized Rayleigh-G Family. Within this family, we present a special sub-model known as the odd Generalized Rayleigh Inverse Weibull (OGRIW) distribution. The OGRIW distribution is derived by combining the T-X family and the Generalized Rayleigh distribution. We provide a comprehensive expansion of the (PDF) and (CDF) for the OGRIW distribution. Additionally, we investigate several mathematical properties of the OGRIW distribution, including moments, moment-generating function, incomplete moments, quantile function, order statistics and Rényi entropy. To estimate the model parameters, we employ the maximum likelihood method, aiming to identify the parameter values that maximise the likelihood of the observed data. Finally, we apply the proposed OGRIW distribution to two real COVID-19 datasets from Mexico and Canada. The results of these applications demonstrate that the new distribution exhibits remarkable flexibility and outperforms other comparative distributions in terms of accurately modelling the COVID-19 data.

A generating family of unit-Garima distribution: Properties, likelihood inference, and application

This article proposes the unit Garima (UGa) distribution for analysing proportion data. Some statistical properties of the UGa distribution are investigated, including survival and hazard functions, order statistics, quantile function, and stress-strength reliability measure. Next, a new family of continuous distributions, called the unit Garima-generated (UGa-G) family of distributions, is studied. The UGa-G family of distributions has the feature to use the UGa distribution as the main generator and the concept of the T-X family of distributions. Some UGa-G family sub-models are provided, such as the UGa-Beta, UGa-Weibull, and UGa-normal distributions. The maximum likelihood method is used to estimate the model parameters for the statistical aspect. A Monte Carlo simulation for the percentile bootstrap confidence intervals for each parameter of the proposed distributions is provided. Applications to eight practical data sets are given to demonstrate the usefulness of the proposed distributions.

A Family of Truncated Positive Distributions

In this paper, a new family of continuous distributions with positive support is introduced. This family is generated by a truncation of the family of univariate symmetrical distributions. In this new family of distributions, general properties, such as moments, asymmetry and kurtosis coefficients, are derived. Particular cases of interest based on the normal, logistic, Laplace and Cauchy models are discussed in depth. The estimation of parameters is carried out by applying moments and maximum likelihood methods. Also, a simulation study was conducted to illustrate the good performance of estimators. An application to the Survival Times (in days) of Guinea Pigs dataset is included, where the special cases of distributions in this family are fitted. The option which provides the best fit is ultimately chosen. An R package, called “tpn”, has been implemented, which includes the relevant cases of interest in this family.

The Type II Quasi Lambert Family

Probability distributions and their families play an effective role in statistical modeling and statistical analysis. Recently, researchers have been increasingly interested in generating new families with high flexibility and low number of milestones. We propose and study a new family of continuous distributions. Relevant properties are presented. Many bivariate versions of the new family are derived under the Farlie-Gumbel-Morgenstern copula, modified Farlie-Gumbel-Morgenstern copula, Clayton copula, entropy copula and Ali-Mikhail-Haq copula. We present two characterizations of the new family. Different estimation methods such as the maximum likelihood estimation, maximum product spacing estimation, least squares estimation, weighted least squares estimation, Anderson-Darling estimation and the Cramer-von Mises estimation methods are considered. Simulation studies for comparing estimation methods are performed based on the baseline Lindley model. Two real data sets are analyzed for comparing the competitive models.

A weighted Topp-Leone G family of distributions: properties, applications for modelling reliability data and different method of estimation

Based on the Topp-Leone distribution, we propose a new family of continuous distributions with one shape parameter called the weighted Topp-Leone family. We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics.Then we study Lindley case as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, three real data sets are illustration the purposes.

The type 2 extended exponentiated family of distributions

In this work, we propose a new family of continuous distributions with two shape parameters named the Extended Exponentiated type 2 (EE2-H). We study some basic properties including quantile function, mixture for cdf and pdf and order statistics .Then we study half-logistic distribution as a special case with more details. We study different methods of estimation and then we compare these methods with a simulation study. Finally, two real data sets show the capability of this model.

The Gudermannian Generated Family of Distributions with Characterizations, Regression Models and Applications

This study proposes a new family of continuous distributions, called the Gudermannian generated family of distributions, based on the Gudermannian function. The statistical properties, including moments, incomplete moments and generating functions, are studied in detail. Simulation studies are performed to discuss and evaluate the maximum likelihood estimations of the model parameters. The regression model of the proposed family considering the heteroscedastic structure of the scale parameter is defined. Three applications on real data sets are implemented to convince the readers in favour of the proposed models.

An extended type I half-logistic family of distributions: Properties, applications and different method of estimations

Abstract We propose a new family of continuous distributions with two shape parameters called the Extended type I(ET1HL-G). We study some basic properties including quantile function, asymptotic, mixture for cdf and pdf, various entropies and order statistics. Then we study half-logistic distribution as special case with more details. The maximum likelihood estimates of parameters are compared with various methods of estimations by conducting a simulation study. Finally, two data sets are illustration the purposes.

Generalization method of generating the continuous nested distributions

Abstract In many life time scenarios, life of one component or system nested in other components or systems. To model these complex structures some so called nested models are required rather than conventional models. This paper introduces the generalization of the method of generating continuous distribution proposed by N. Eugene, C. Lee, and F. Famoye, “Beta-normal distribution and its applications,” Commun. Stat. Theor. Methods, vol. 31, no. 4, pp. 497–512, 2002 and A. Alzaatreh, C. Lee, and F. Famoye, “A new method for generating families of continuous distributions,” Metron, vol. 71, no. 1, pp. 63–79, 2013 which nest one model in other to cope with complex systems. Some important characteristics of the proposed family of generalized distribution have been studied. The famous Beta, Kumaraswami and Gamma generated distributions are special cases of our suggested procedure. Some new distributions have also been developed by using the suggested methodology and their important properties have been discussed as well. A variety of real life data sets are used to demonstrate the efficacy of new suggested distributions and illation is made with baseline models.

A New Family of Continuous Distributions: Properties and Estimation

We introduce a new flexible modified alpha power (MAP) family of distributions by adding two parameters to a baseline model. Some of its mathematical properties are addressed. We show empirically that the new family is a good competitor to the Beta-F and Kumaraswamy-F classes, which have been widely applied in several areas. A new extension of the exponential distribution, called the modified alpha power exponential (MAPE) distribution, is defined by applying the MAP transformation to the exponential distribution. Some properties and maximum likelihood estimates are provided for this distribution. We analyze three real datasets to compare the flexibility of the MAPE distribution to the exponential, Weibull, Marshall–Olkin exponential and alpha power exponential distributions.

Type I Half Logistic Burr X-G Family: Properties, Bayesian, and Non-Bayesian Estimation under Censored Samples and Applications to COVID-19 Data

In this paper, we present a new family of continuous distributions known as the type I half logistic Burr X-G. The proposed family’s essential mathematical properties, such as quantile function (QuFu), moments (Mo), incomplete moments (InMo), mean deviation (MeD), Lorenz (Lo) and Bonferroni (Bo) curves, and entropy (En), are provided. Special models of the family are presented, including type I half logistic Burr X-Lomax, type I half logistic Burr X-Rayleigh, and type I half logistic Burr X-exponential. The maximum likelihood (MLL) and Bayesian techniques are utilized to produce parameter estimators for the recommended family using type II censored data. Monte Carlo simulation is used to evaluate the accuracy of estimates for one of the family’s special models. The COVID-19 real datasets from Italy, Canada, and Belgium are analysed to demonstrate the significance and flexibility of some new distributions from the family.

Robust Assessing the Lifetime Performance of Products with Inverse Gaussian Distribution in Bayesian and Classical Setup

The inverse Gaussian (Wald) distribution belongs to the two-parameter family of continuous distributions having a range from 0 to ∞ and is considered as a potential candidate to model diffusion processes and lifetime datasets. Bayesian analysis is a modern inferential technique in which we estimate the parameters of the posterior distribution obtained by formally combining a prior distribution with an observed data distribution. In this article, we have attempted to perform the Bayesian and classical analyses of the Wald distribution and compare the results. Jeffreys' and uniform priors are used as noninformative priors, while the exponential distribution is used as an informative prior. The analysis comprises finding joint posterior distributions, the posterior means, predictive distributions, and credible intervals. To illustrate the entire estimation procedure, we have used real and simulated datasets, and the results thus obtained are discussed and compared. We have used the Bayesian specialized Open BUGS software to perform Markov Chain Monte Carlo (MCMC) simulations using a real dataset.

A New Family of Continuous Distributions: Properties, Copulas and Real Life Data Modeling

A new family of distributions called the Kumaraswamy Rayleigh family is defied and studied. Some of its relevant statistical properties are derived. Many new bivariate type G families using the of Farlie-Gumbel-Morgenstern, modified Farlie-Gumbel-Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. The method of the maximum likelihood estimation is used. Some special models based on log-logistic, exponential, Weibull, Rayleigh, Pareto type II and Burr type X, Lindley distributions are presented and studied. Three dimensional skewness and kurtosis plots are presented. A graphical assessment is performed. Two real life applications to illustrate the flexibility, potentiality and importance of the new family is proposed.

A New Modified Kies Family: Properties, Estimation Under Complete and Type-II Censored Samples, and Engineering Applications

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.

The Poisson exponential-G family of distributions with properties and applications

We introduce a new family of continuous distributions called the Poisson-exponential-G family based on the Poisson-exponential distribution of Cancho et al. (2011). The shapes of the density and hazard functions corresponding to new family are studied. Two sub-models are also discussed. We study some properties of the proposed family such as moments, generating functions, entropy measures, stress-strength model and order statistics. The model parameters are estimated via maximum likelihood approach. A Monte-Carlo simulation study is designed and used to show the performances of obtained estimators of parameters. Two real applications are introduced to show the importance of the new family.

Class enumeration false positive in skew-t family of continuous growth mixture models.

Growth Mixture Modeling (GMM) has gained great popularity in the last decades as a methodology for longitudinal data analysis. The usual assumption of normally distributed repeated measures has been shown as problematic in real-life data applications. Namely, performing normal GMM on data that is even slightly skewed can lead to an over selection of the number of latent classes. In order to ameliorate this unwanted result, GMM based on the skew t family of continuous distributions has been proposed. This family of distributions includes the normal, skew normal, t, and skew t. This simulation study aims to determine the efficiency of selecting the “true” number of latent groups in GMM based on the skew t family of continuous distributions, using fit indices and likelihood ratio tests. Results show that the skew t GMM was the only model considered that showed fit indices and LRT false positive rates under the 0.05 cutoff value across sample sizes and for normal, and skewed and kurtic data. Simulation results are corroborated by a real educational data application example. These findings favor the development of practical guides of the benefits and risks of using the GMM based on this family of distributions.

Regression models for exceedance data: a new approach

The generalized Pareto distribution (GPD) is a family of continuous distributions used to model the tail of the distribution to values higher than a threshold u. Despite the advantages of the GPD representation, its shape and scale parameters do not correspond to the expected value, which complicates the interpretation of regression models specified using the GPD. This study proposes a linear regression model in which the response variable is a GPD, using a new parametrization that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive real line response variable, as is usual in the context of generalized linear models. Furthermore, we propose a model for extreme values, in which the GPD parameters (mean and precision) are defined on the basis of a dynamic linear regression model. The novelty of the study lies in the time variation of the mean and precision parameter of the resulting distribution. The parameter estimation of these new models is performed under the Bayesian paradigm. Simulations are conducted to analyze the performance of our proposed models. Finally, the models are applied to environmental datasets (temperature datasets), illustrating their capabilities in challenging cases in extreme value theory.

Modified Odd Weibull Family of Distributions: Properties and Applications

In this paper, a new family of continuous distributions, called the modified odd Weibull-G (MOW-G) family, is studied. The MOW-G family has the feature to use the Weibull distribution as main generator and a new modification of the odd transformation, opening new horizon in terms of statistical modelling. Its main theoretical and practical aspects are explored. In particular, for the mathematical properties, we investigate some results in distribution, quantile function, skewness, kurtosis, moments, moment generating function, order statistics and entropy. For the statistical aspect, the maximum likelihood estimation method is used to estimate the model parameters. The performance of this method is evaluated by a Monte Carlo simulation study. Applications to three practical data sets are given to demonstrate the usefulness of the MOW-G model.

Type II Power Topp-Leone Generated Family of Distributions with Statistical Inference and Applications

In this paper, we present and study a new family of continuous distributions, called the type II power Topp-Leone-G family. It provides a natural extension of the so-called type II Topp-Leone-G family, thanks to the use of an additional shape parameter. We determine the main properties of the new family, showing how they depend on the involving parameters. The following points are investigated: shapes and asymptotes of some important functions, quantile function, some mixture representations, moments and derivations, stochastic ordering, reliability and order statistics. Then, a special model of the family based on the inverse exponential distribution is introduced. It is of particular interest because the related probability functions are tractable and possess various kinds of asymmetric shapes. Specially, reverse J, left skewed, near symmetrical and right skewed shapes are observed for the corresponding probability density function. The estimation of the model parameters is performed by the use of three different methods. A complete simulation study is proposed to illustrate their numerical efficiency. The considered model is also applied to analyze two different kinds of data sets. We show that it outperforms other well-known models defined with the same baseline distribution, proving its high level of adaptability in the context of data analysis.

The Fréchet Topp Leone-G Family of Distributions: Properties, Characterizations and Applications

A new family of continuous distributions which ensure model flexiblity, is introduced based on the Frechet distribution and Topp Leone-G family. Two special sub-models of the new family are discussed. We provide some distributional properties of this family in the general setting such as the series expansions of density, moments, generating function, stress strength model, Renyi and Shannon entropies, probability weighted moments and order statistics. Certain characterizations of the proposed family are presented. The maximum likelihood estimates and the observed information matrix are obtained for the model parameters. We assess the performance of the maximum likelihood estimators by means of a graphical simulation study. The potentiality of the new class is shown via two applications to real data sets.