On the new modified Dagum distribution: Properties and applications

A new univariate extension of the Fréchet distribution is proposed and studied. Some of its fundamental statistical properties such as stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction using Morgenstern family and via Clayton Copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown by means of two applications to real data sets.

<h2>Abstract</h2>\n<p><strong>Objective:</strong> To find appropriate model more flexible than classical models for fitting the survival data in health research especially in cancer blood (Leukemia) studies by introducing some additional parameters to the basic models.<strong> Methods:</strong> Based on the Alpha Power transformation technique and using Two-Parameter Odoma distribution, the new distribution called the Alpha Power Two- Parameter Odoma distribution is introduced and studied. The maximum likelihood estimation procedure is employed to estimate the unknown parameters. A simulation study is carried out to evaluate the performance of the maximum likelihood estimators. <strong>Finding:</strong> Common statistical properties such as quantile function, rth moments, moment generating function, characteristic function, incomplete moments, entropy and order statistics are derived. The practical importance of the proposed model is illustrated by using goodness-of-fit criterias based on real data set.  The APTPO distribution is the most appropriate model for fitting survival data <strong>Novelty:</strong>of cancer studies comparing with other competitive distributions.</p>\n<p><strong>Keywords:</strong> TwoParameter Odoma Distribution; Alpha Power Transformation; Quantile; Moments; Maximum Likelihood Estimation</p>

A new compound extension of the Fréchet distribution is introduced and studied. Some of its properties including moments, incomplete moments, probability weighted moments, moment generating function, stress strength reliability model, residual life and reversed residual life functions are derived. The mean squared errors (MSEs) for some estimation methods including maximum likelihood estimation (MLE), Cram\'{e}r--von Mises (CVM) estimation, Bootstrapping (Boot.) estimation and Kolmogorov estimates (KE) method are used to estimate the unknown parameter via a simulation study. Two real applications are presented for comparing the estimation methods. Another two real applications are presented for comparing the competitive models. The nonparametric Hill estimator under the breaking stress of carbon fibers is estimated using the tail index (TIx) of the new model. Finally, a case study on reliability analysis of composite materials for aerospace applications is presented.

In this paper, we consider Dagum distribution which is capable of modeling various shapes of failure rates and aging criteria. Based on progressively type-I interval censoring data, we first obtain the maximum likelihood estimators and the approximate confidence intervals of the unknown parameters of the Dagum distribution. Next, we obtain the Bayes estimators of the parameters of Dagum distribution under the squared error loss (SEL) and balanced squared error loss (BSEL) functions using independent informative gamma and non informative uniform priors for both scale and two shape parameters. A Monte Carlo simulation study is performed to assess the performance of the proposed Bayes estimators with the maximum likelihood estimators. We also compute credible intervals and symmetric 100(1 − τ)% two-sided Bayes probability intervals under the respective approaches. Besides, based on observed samples, Bayes predictive estimates and intervals are obtained using one-and two-sample schemes. Simulation results reveal that the Bayes estimates based on SEL and BSEL performs better than maximum likelihood estimates in terms of bias and MSEs. Besides, credible intervals have smaller interval lengths than confidence interval. Further, predictive estimates based on SEL with informative prior performs better than non-informative prior for both one and two sample schemes. Further, the optimal censoring scheme has been suggested using a optimality criteria. Finally, we analyze a data set to illustrate the results derived.

The transmuted family of distributions has been receiving increased attention over the last few years. In this paper, we generalize the Marshall-Olkin extended Lomax distribution using the quadratic rank transmutation map to obtain the transmuted Marshall-Olkin extended Lomax distribution. Several properties of the new distribution are discussed including the hazard rate function, ordinary and incomplete moments, characteristic function and order statistics. We provide an estimation procedure by the maximum likelihood method and a simulation study to assess the performance of the new distribution. We prove empirically the flexibility of the new model by means of an application to a real data set. It is superior to other three and four parameter lifetime distributions.

This paper introduces a new generalization of the transmuted Marshall-Olkin Frechet distribution of Afify et al. (2015), using Kumaraswamy generalized family. The new model is referred to as Kumaraswamy transmuted Marshall-Olkin FrØchet distribution. This model contains sixty two sub-models as special cases such as the Kumaraswamy transmuted Frechet, Kumaraswamy transmuted Marshall-Olkin, generalized inverse Weibull and Kumaraswamy Gumbel type II distributions, among others. Various mathematical properties of the proposed distribution including closed forms for ordinary and incomplete moments, quantile and generating functions and Renyi and -entropies are derived. The unknown parameters of the new distribution are estimated using the maximum likelihood estimation. We illustrate the importance of the new model by means of two applications to real data sets.

This paper proposes a new generator function based on the inverted Kumaraswamy distribution and introduces ‘generalized inverted Kumaraswamy-G’ family of distributions. We provide a comprehensive account of some of its mathematical properties that include the ordinary and incomplete moments, quantile and generating functions and order statistics. The infinite mixture representations for probability density and cumulative distribution and entropy functions of the new family are also established. The density function of the ith-order statistics is expressed as an infinite linear combination of baseline densities and model parameters are estimated by maximum likelihood method. Four special models of this family are also derived along with their respective hazard rate functions. The maximum likelihood estimation (MLE) method is used to obtain the model parameters. Monte Carlo simulation experiments are executed to assess the performance of the ML estimators under the corresponding generated models while some data applications are also illustrated. The results of the study show that the proposed distribution is more flexible as compared to the baseline model. This distribution especially can be used to model symmetric, left-skewed, right-skewed and reversed-J data sets.

In this work, we introduce a novel generalization of the extended exponential distribution with four parameters through the Kumaraswamy family. The proposed model is referred to as the Kumaraswamy extended exponential (KwEE). The significance of the suggested distribution from its flexibility in applications and data modeling. As specific sub-models, it includes the exponential, Kumaraswamy exponential, Kumaraswamy Lindley, Lindley, extended exponential, exponentiated Lindley, gamma and generalized exponential distributions. The representation of the density function, quantile function, ordinary and incomplete moments, generating function, and reliability of the KwEE distribution are all derived. The maximum likelihood approach is used to estimate model parameters. A simulation study for maximum likelihood estimates was used to investigate the behaviour of the model parameters. A numerical analysis is performed for various sample sizes and parameter values to analyze the behaviour of estimates using accuracy measures. According to a simulated investigation, the KwEE's maximum likelihood estimates perform well with increased sample size. We provide two real-world examples utilizing applied research to demonstrate that the new model is more effective.

We introduce a new generalized family of nonnegative continuous distributions by adding two extra parameters to a lifetime distribution, called the baseline distribution, by twice compounding a power series distribution. The new family, called the lifetime power series-power series family, has a serial arrangement of parallel structures, which extends the Marshall and Olkin structure. Four special models are discussed. A mathematical treatment of the new distributions is provided, including ordinary and incomplete moments, quantile, moment generating and mean residual functions. The maximum likelihood estimation technique is used to estimate the model parameters and a simulation study is conducted to investigate the performance of the maximum likelihood estimates. Its applicability is also illustrated by means of two real data sets.

We propose and study a new five-parameter continuous distribution in the unit interval through a specific probability integral transform. The new distribution, under some parameter constraints, is an identified parametric model that includes as special cases six important models such as the Kumaraswamy and beta distributions. We obtain ordinary and incomplete moments, quantile and generating functions, mean deviations, R\'enyi entropy and moments of order statistics. The estimation of the model parameters is performed by maximum likelihood, and hypothesis tests are discussed. Additionally, through a simulation study we investigate the behavior of the maximum likelihood estimator, also we investigate the impact of ignoring identifiability problems. The usefulness of the proposed distribution is illustrated by means of a real data set.

We propose and develop the four-parameter Harris Extended Fréchet distribution. It is obtained by inserting the two-parameter Frechet distribution as the baseline in the Harris family and may be a useful alternative method to model income distribution and could be applied to other areas. We demonstrate that the new distribution can have decreasing, increasing and upside-down-bathtub hazard functions and that its probability density function is an infinite linear combination of Frechet densities. Some standard mathematical properties of the proposed distribution are derived, such as the quantile function, ordinary and incomplete moments, incomplete moments, Lorenz and Bonferroni curves, Gini index, Renyi and ????-entropies, mean residual life and mean inactivity time, probability weighted moments, stress-strength reliability, and order statistics. We also obtain the maximum likelihood estimators of the model. The potentiality/flexibility of the new distribution is illustrated by means two applications to failure and waiting time data sets

In this paper, a new three-parameter lifetime model, called the odd log-logistic generalised Lindley distribution, is introduced. Some structural properties of the new distribution including ordinary and incomplete moments, quantile and generating functions and order statistics are obtained. The new density function can be expressed as a linear mixture of exponetiated Lindley densities. Different methods are discussed to estimate the model parameters and a simulation study is carried out to show the performance of the new distribution. The importance and flexibility of the new model are also illustrated empirically by means of two real data sets. Finally, Bayesian analysis and Gibbs sampling are performed based on the two real data sets.

In this paper, a new three-parameter lifetime model, called the new odd log-logistic Lindley (NOLL-L) distribution, is introduced. Some structural properties of the new distribution including ordinary and incomplete moments, quantile and generating functions and order statistics are obtained. The new density function can be expressed as a linear mixture of exponentiated Lindley densities. The different methods are discussed to estimate the model parameters and a simulation study is done to show the performance of the new distribution. The importance and flexibility of the new model are also illustrated empirically by means of two real data sets.

We consider goodness-of-fit tests of the Cauchy distribution based on weighted integrals of the squared distance between the empirical characteristic function of the standardized data and the characteristic function of the standard Cauchy distribution. For standardization of data Gurtler and Henze (2000,Annals of the Institute of Statistical Mathematics,52, 267–286) used the median and the interquartile range. In this paper we use the maximum likelihood estimator (MLE) and an equivariant integrated squared error estimator (EISE), which minimizes the weighted integral. We derive an explicit form of the asymptotic covariance function of the characteristic function process with parameters estimated by the MLE or the EISE. The eigenvalues of the covariance function are numerically evaluated and the asymptotic distributions of the test statistics are obtained by the residue theorem. A simulation study shows that the proposed tests compare well to tests proposed by Gurtler and Henze and more traditional tests based on the empirical distribution function.

The article is concerned with parameter estimation for Gaussian Markov random fields observed with additive independent identically distributed noise. In particular, we consider maximum likelihood and maximum pseudo-likelihood estimation for the noise-free case and make adjustments to the estimators in the presence of noise. The adjusted maximum likelihood estimator is computed in O(n2) time for a square image with n pixels. The estimation method is useful when only the moments of the noise are specified or when the exact maximum likelihood estimator is difficult to compute (e.g., for certain non-Gaussian noise distributions). The adjusted maximum pseudo-likelihood estimator is straightforward to calculate, is useful as a starting value in maximization routines, and is often a reasonable estimator in its own right. We discuss asymptotic properties of the adjusted estimators including consistency. We also consider constrained maximum pseudo-likelihood estimation and a Bayesian estimator. We compare the adjusted estimators with the exact Gaussian maximum likelihood estimator and toroidal boundary approximation Gaussian maximum likelihood estimator in a simulation study. The adjusted estimators are also robust to the specification of the noise distribution.