Incomplete Moments Research Articles

Bayesian and Frequentist Inferences on a Type I Half-Logistic Odd Weibull Generator with Applications in Engineering.

In this article, we have proposed a new generalization of the odd Weibull-G family by consolidating two notable families of distributions. We have derived various mathematical properties of the proposed family, including quantile function, skewness, kurtosis, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves, probability weighted moments, moments of (reversed) residual lifetime, entropy and order statistics. After producing the general class, two of the corresponding parametric statistical models are outlined. The hazard rate function of the sub-models can take a variety of shapes such as increasing, decreasing, unimodal, and Bathtub shaped, for different values of the parameters. Furthermore, the sub-models of the introduced family are also capable of modelling symmetric and skewed data. The parameter estimation of the special models are discussed by numerous methods, namely, the maximum likelihood, simple least squares, weighted least squares, Cramér-von Mises, and Bayesian estimation. Under the Bayesian framework, we have used informative and non-informative priors to obtain Bayes estimates of unknown parameters with the squared error and generalized entropy loss functions. An extensive Monte Carlo simulation is conducted to assess the effectiveness of these estimation techniques. The applicability of two sub-models of the proposed family is illustrated by means of two real data sets.

The Kummer Beta Normal: A New Useful-Skew Model

The normal distribution is the most popular model in applications to real data. We propose a new extension of this distribution, called the Kummer beta normal distribution, which presents greater flexibility to model scenarios involving skewed data. The new probability density function can be represented as a linear combination of exponentiated normal pdfs. We also propose analytical expressions for some mathematical quantities: Ordinary and incomplete moments, mean deviations and order statistics. The estimation of parameters is approached by the method of maximum likelihood and Bayesian analysis. Likelihood ratio statistics and formal goodnessof-fit tests are used to compare the proposed distribution with some of its sub-models and non-nested models. A real data set is used to illustrate the importance of the proposed model.

A new generalization of the inverse Lomax distribution with statistical properties and applications

In this paper, we introduce a new generalization of the inverse Lomax distribution with one extra shape parameter, the so-called power inverse Lomax (PIL) distribution, derived by using the power transformation method. We provide a more flexible density function with right-skewed, uni-modal, and reversed J-shapes. The new three-parameter lifetime distribution capable of modeling decreasing, Reversed-J and upside-down hazard rates shapes. Some statistical properties of the PIL distribution are explored, such as quantile measure, moments, moment generating function, incomplete moments, residual life function, and entropy measure. The estimation of the model parameters is discussed using maximum likelihood, least squares, and weighted least squares methods. A simulation study is carried out to compare the efficiencies of different methods of estimation. This study indicated that the maximum likelihood estimates are more efficient than the corresponding least squares and weighted least squares estimates in approximately most of the situations Also, the mean square errors for all estimates are decreasing as the sample size increases. Further, two real data applications are provided in order to examine the flexibility of the PIL model by comparing it with some known distributions. The PIL model offers a more flexible distribution for modeling lifetime data and provides better fits than other models such as inverse Lomax, inverse Weibull, and generalized inverse Weibull.

Lehmann Type II Frechet Poisson Distribution: Properties, Inference and Applications as a Life Time Distribution

A new generalization of the Frechet distribution called Lehmann Type II Frechet Poisson distribution is defined and studied. Various structural mathematical properties of the proposed model including ordinary moments, incomplete moments, generating functions, order statistics, Renyi entropy, stochastic ordering, Bonferroni and Lorenz curve, mean and median deviation, stress-strength parameter are investigated. The maximum likelihood method is used to estimate the model parameters. We examine the performance of the maximum likelihood method by means of a numerical simulation study. The new distribution is applied for modeling three real data sets to illustrate empirically its flexibility and tractability in modeling life time data.

The Kummer Beta Generalized Gamma Distribution

A new extension of the generalized gamma distribution with six parameter called the Kummer beta generalized gamma distribution is introduced and studied. It contains at least 28 special models such as the beta generalized gamma, beta Weibull, beta exponential, generalized gamma, Weibull and gamma distributions and thus could be a better model for analyzing positive skewed data. The new density function can be expressed as a linear combination of generalized gamma densities. Various mathematical properties of the new distribution including explicit expressions for the ordinary and incomplete moments, generating function, mean deviations, entropy, density function of the order statistics and their moments are derived. The elements of the observed information matrix are provided. We discuss the method of maximum likelihood and a Bayesian approach to fit the model parameters. The superiority of the new model is illustrated by means of three real data sets.

The Weibull-G Family of Probability Distributions

The Weibull distribution is the most important distribution for problems in reliability. We study some mathematical properties of the new wider Weibull-G family of distributions. Some special models in the new family are discussed. The properties derived hold to any distribution in this family. We obtain general explicit expressions for the quantile function, ordinary and incomplete moments, generating function and order statistics. We discuss the estimation of the model parameters by maximum likelihood and illustrate the potentiality of the extended family with two applications to real data.

The Kumaraswamy Transmuted-G Family of Distributions: Properties and Applications

We introduce a new class of continuous distributions called the Kumaraswamy transmuted-G family which extends the transmuted class defined by Shaw and Buckley (2007). Some special models of the new family are provided. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi and Shannon entropies, order statistics and probability weighted moments are derived. The maximum likelihood is used for estimating the model parameters. The flexibility of the generated family is illustrated by means of two applications to real data sets.

The Generalized Odd Generalized Exponential Family of Distributions: Properties, Characterizations and Application

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, R𝑒́nyi, Shannon and q-entropies, order statistics and probability weighted moments are derived. We also propose bivariate generalizations. We constructed a simple type Copula and intro-duced 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.

The exponentiated generalized extended exponential distribution

We introduce and study a new four-parameter lifetime model named the exponentiated generalized extended exponential distribution. The proposed model has the advantage of including as special cases the exponential and exponentiated exponential distributions, among others, and its hazard function can take the classic shapes: bathtub, inverted bathtub, increasing, decreasing and constant, among others. We derive some mathematical properties of the new model such as a representation for the density function as a double mixture of Erlang densities, explicit expressions for the quantile function, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R´enyi entropy, density of order statistics and reliability. We use the maximum likelihood method to estimate the model parameters. Two applications to real data illustrate the flexibility of the proposed model.

THE WEIBULL GENERALIZED FLEXIBLE WEIBULL EXTENSION DISTRIBUTION

This paper introduces a new four parameters model called the Weibull Generalized Flexible Weibull extension (WGFWE) distribution which exhibits bathtub-shaped hazard rate. Some of it's statistical properties are obtained including ordinary and incomplete moments, quantile and generating functions, reliability and order statistics. The method of maximum likelihood is used for estimating the model parameters and the observed Fisher's information matrix is derived. We illustrate the usefulness of the proposed model by applications to real data.

An Extended Fr echet Distribution: Properties and Applications

In this paper, we introduce an extended four-parameter Fr´echet model called the exponentiated exponential Fr´echet distribution, which arises from the quantile function of the standard exponential distribution. Various of its mathematical properties are derived including the quantile function, ordinary and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, generating function, Shannon entropy and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The usefulness of the new distribution is illustrated by means of three real lifetime data sets. In fact, the new model provides a better fit to these data than the Marshall-Olkin Fr´echet, exponentiated-Fr´echet and Fr´echet models.

The Odd Lindley Burr XII Model: Bayesian Analysis, Classical Inference and Characterizations

In this work, we study the odd Lindley Burr XII model initially introduced by Silva et al. [29]. This model has the advantage of being capable of modeling various shapes of aging and failure criteria. Some of its statistical structural properties including ordinary and incomplete moments, quantile and generating function and order statistics are derived. The odd Lindley Burr XII density can be expressed as a simple linear mixture of BurrXII densities. Useful characterizations are presented. The maximum likelihood method is used to estimate the model parameters. Simulation results to assess the performance of the maximum likelihood estimators are discussed. We prove empirically the importance and flexibility of the new model in modeling various types of data. Bayesian estimation is performed by obtaining the posterior marginal distributions as well as using the simulation method of Markov Chain Monte Carlo (MCMC) by the Metropolis-Hastings algorithm in each step of Gibbs algorithm. The trace plots and estimated conditional posterior distributions are also presented.

New Modified Singh-Maddala Distribution: Development, Properties, Characterizations And Applications

In this paper, a new five-parameter extended Burr XII model called new modified Singh-Maddala (NMSM) is developed from cumulative hazard function of the modified log extended integrated beta hazard (MLEIBH) model. The NMSM density function is left-skewed, right-skewed and symmetrical. The Lambert W function is used to study descriptive measures based on quantile, moments, and moments of order statistics, incomplete moments, inequality measures and residual life function. Different reliability and uncertainty measures are also theoretically established. The NMSM distribution is characterized via different techniques and its parameters are estimated using maximum likelihood method. The simulation studies are performed on the basis of graphical results to illustrate the performance of maximum likelihood estimates (MLEs) of the parameters. The significance and flexibility of NMSM distribution is tested through different measures by application to two real data sets.

The Transmuted Generalized Odd Generalized Exponential-G Family of Distributions: Theory and Applications

We propose a new generator of continuous distributions, so called the transmuted generalized odd generalized exponential-G family, which extends the generalized odd generalized exponential-G family introduced by Alizadeh et al. (2017). Some statistical properties of the new family such as; raw and incomplete moments, moment generating function, Lorenz and Bonferroni curves, probability weighted moments, Rényi entropy, stress strength model and order statistics are investigated. The parameters of the new family are estimated by using the method of maximum likelihood. Two real applications are presented to demonstrate the effectiveness of the suggested family.

Type II Generalized Topp–Leone Family of Distributions: Properties and Applications

The Topp-Leone distribution is an attractive model for life testing and reliability studies as it acquires a bathtub shaped hazard function. In this paper, we introduce a new family of distributions, depending on Topp–Leone random variable as a generator, called the Type II generalized Topp– Leone–G (TIIGTL-G) family. Its density function can be unimodel, leftskewed, right-skewed, and reversed-J shaped, and has increasing, decreasing, upside-down, J and reversed-J hazard rates. Some special models are presented. Some of its statistical properties are studied. Explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Rényi entropy and order statistics are derived. The method of maximum likelihood is used to estimate the model parameters. The importance of one special model; namely; the Type II generalized Topp–Leone exponential is illustrated through two real data sets.

A New Distribution for Extreme Values: Regression Model, Characterizations and Applications

A new four parameter extreme value distribution is defined and studied. Various structural properties of the proposed distribution including ordinary and incomplete moments, generating functions, residual and reversed residual life functions, order statistics are investigated. Some useful characterizations based on two truncated moments as well as based on the reverse hazard function and on certain functions of the random variable are presented. The maximum likelihood method is used to estimate the model parameters. Further, we propose a new extended regression model based on the logarithm of the new distribution. The new distribution is applied to model three real data sets to prove empirically its flexibility.

Properties of the Transmuted Burr XII Distribution, Regression and its Applications

In this paper, we define and study a four-parameter model called the transmuted Burr XII distribution. We obtain some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, order statistics, probability weighted moments and entropies. We formulate and develop a log-linear model using the new distribution so-called the log-transmuted Burr XII distribution for modeling data with a unimodal failure rate function, as an alternative to the log-McDonald Burr XII, log-beta Burr XII, log-Kumaraswamy Burr XII, log-Burr XII and logistic regression models. The flexibility of the proposed models is illustrated by means of three applications to real data sets.

The Inverse Weibull Generator of Distributions: Properties and Applications

In this paper, we introduce a new family of univariate distributions with two extra positive parameters generated from inverse Weibull random variable called the inverse Weibull generated (IW-G) family. The new family provides a lot of new models as well as contains two new families as special cases. We explore four special models for the new family. Some mathematical properties of the new family including quantile function, ordinary and incomplete moments, probability weighted moments, Rѐnyi entropy and order statistics are derived. The estimation of the model parameters is performed via maximum likelihood method. Applications show that the new family of distributions can provide a better fit than several existing lifetime models.

A New Generalized Family of Distributions from Bounded Support

In this paper, we introduce a new generalized family of distri- butions from bounded support (0,1), namely, the Topp-Leone-G family.Some of mathematical properties of the proposed family have been studied. The new density function can be symmetrical, left-skewed, right-skewed or reverse-J shaped. Furthermore, the hazard rate function can be constant, in- creasing, decreasing, J or bathtub hazard rate shapes. Three special models are discussed. We obtain simple expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations and entropies. The method of maximum likelihood is used to estimate the model parame- ters. The flexibility of the new family is illustrated by means of three real data sets.

A new one-parameter lifetime distribution and its regression model with applications.

Lifetime distributions are an important statistical tools to model the different characteristics of lifetime data sets. The statistical literature contains very sophisticated distributions to analyze these kind of data sets. However, these distributions have many parameters which cause a problem in estimation step. To open a new opportunity in modeling these kind of data sets, we propose a new extension of half-logistic distribution by using the odd Lindley-G family of distributions. The proposed distribution has only one parameter and simple mathematical forms. The statistical properties of the proposed distributions, including complete and incomplete moments, quantile function and Rényi entropy, are studied in detail. The unknown model parameter is estimated by using the different estimation methods, namely, maximum likelihood, least square, weighted least square and Cramer-von Mises. The extensive simulation study is given to compare the finite sample performance of parameter estimation methods based on the complete and progressive Type-II censored samples. Additionally, a new log-location-scale regression model is introduced based on a new distribution. The residual analysis of a new regression model is given comprehensively. To convince the readers in favour of the proposed distribution, three real data sets are analyzed and compared with competitive models. Empirical findings show that the proposed one-parameter lifetime distribution produces better results than the other extensions of half-logistic distribution.