Family Of Generalized Distributions Research Articles

Inference for compound truncated Poisson log-normal model with application to maximum precipitation data

The main objective of this paper is to propose a new general family of distributions, namely compound truncated Poisson log-normal distribution of which log-normal distribution is a special case. The proposed model has three unknown parameters, and it can take variety of shapes. It can be used effectively in analyzing maximum precipitation data during a particular period of time obtained from different stations. It is assumed that the number of stations operate follows a zero-truncated Poisson random variables, and the daily precipitation follows a log-normal random variable. The maximum likelihood estimators can be obtained quite conveniently using Expectation-Maximization (EM) algorithm. Approximate maximum likelihood estimators are also derived. The associated confidence intervals can also be obtained from the observed Fisher information matrix. Simulation results have been performed to check the performance of the EM algorithm, and it is observed that the EM algorithm works quite well in this case. When we analyze the precipitation data set using the proposed model it is observed that the proposed model provides better fit than some of the existing models.

Stochastic comparisons of two finite mixtures of general family of distributions

Stochastic comparisons of two finite mixtures of general family of distributions

Bayesian and non-Bayesian inference for a general family of distributions based on simple step-stress life test using TRV model under type II censoring

In this article, we consider the parametric inference, using Type II censored data, based on the tampered random variable (TRV) model for simple step-stress life testing (SSLT). We have taken the members of the Lehmann family of distributions as the baseline lifetimes of the experimental units under normal stress conditions. Based on Type II censored data and a simple SSLT framework, we obtain the maximum likelihood estimator (MLE) and the Bayes estimators of the model parameters. Further, we obtain asymptotic confidence intervals of the unknown model parameters using the observed Fisher information matrix. Moreover, bootstrap confidence intervals were constructed. The Bayes estimators are computed using the Markov chain Monte Carlo (MCMC) method under the squared error loss function and the LINEX loss function. We also construct the highest posterior density (HPD) credible intervals of the unknown model parameters. Extensive simulation studies are performed to investigate the finite sample properties of the proposed estimators. Finally, the proposed methods are illustrated with the analysis of a real data set.

Order restricted inference for a multiple step‐stress model with long‐term survivors for a general family of distributions

AbstractIn this article, we propose a failure rate based step‐stress accelerated life testing (SSALT) model assuming that the time‐to‐event distribution belongs to a fairly general family of distributions and the underlying population consists of long term survivors. With increase in stress levels, it is expected that the mean time to the event of interest gets shortened leading to an order restriction among the mean times‐to‐event. We propose here a method of obtaining order restricted maximum likelihood estimators (MLEs) based on expectation maximization (EM) algorithm coupled with the method of generalized isotonic regression technique. Additionally, we address the testing of hypothesis problem for the presence of long term survivors in the underlying population based on both asymptotic and non‐asymptotic approaches. To illustrate the effectiveness of the proposed method, extensive simulation experiments are carried out and a real data set is analyzed.

Complete Study of an Original Power-Exponential Transformation Approach for Generalizing Probability Distributions

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are ideal for fitting several real data sets encountered in applied fields. Some members of the MGG family are proposed for special distributions. Following that, the uniform distribution is chosen as a baseline distribution to yield the modified uniform (MU) distribution with the goal of efficiently modeling measures with bounded values. Some useful key properties of the MU distribution are determined. The estimation of the unknown parameters of the MU model is discussed using seven methods, and then, a simulation study is carried out to explore the performance of the estimates. The flexibility of this model is illustrated by the analysis of two real-life data sets. When compared to fair and well-known competitor models in contemporary literature, better-fitting results are obtained for the new model.

A new transmuted family of distributions: Properties and estimation with applications

In this paper, a new transmuted general family of distributions is introduced and studied. It offers an original alternative to the former transmuted family by revisiting its mathematical structure, opening a new perspective on modeling. We derive expressions for the moments, the incomplete moments, the moment generating function, entropy measures, and order statistics as main properties. Further, we introduce a bivariate extension of the new family. Some members of the family are introduced, involving Burr, Gompertz, Weibull and gamma distributions. We discuss the different methods of estimation of the model parameters and a simulation study is performed for one of the special models to check the asymptotic behavior of the estimates under all estimation methods. Further, the interest of the family is illustrated by fitting two real-life data sets to the mentioned sub-models.

Exponentiated odd Lindley-X family with fitting to reliability and medical data sets

This paper concerns constructing a general family of distributions called exponentiated odd Lindley-X (EOL-X) family. We demonstrate that the EOL-X density can be represented as an infinite linear combination of the exponentiated-F densities and, as a result, that many of its mathematical features are derived directly. The fundamental statistical properties, including moments, mean deviations, generating function, order statistics, stochastic ordering, and entropies were investigated. EOL-X special models are introduced. The suggested model presents superior performance when compared to the other models studied, in the reliability and “medical” data. In addition, its bimodal density shape enhances the possibility of good tuning in applications in several other areas, such as survival. Thus, it is expected that this proposal will be of great help to the community studying new families and their adjustments to real data sets.

Alpha Power Odd Generalized Exponential Family of Distributions: Model, Properties and Applications

In this paper, we exhibit a general family of distributions called alpha power odd generalized exponential family of distributions. The new family is very flexible with increasing, decreasing, J, reversed-J, bathtub shapes. Statistical properties of the family such as quantile, expansion of density function, moments, incomplete moments, mean deviation, Bonferroni and Lorenz curves are proposed. The method of maximum likelihood to estimate the model parameters is used. Three applications based on real data sets are the importance and flexibility of the three proposed models.

Some results on stochastic comparisons of two finite mixture models with general components

Finite mixture (FM) models have found key applications in many fields. Recently, some discussions have been made on comparing finite mixture models. In this paper, we discuss stochastic comparison of two FM models with respect to usual stochastic order when the mixture components have a general family of distributions. This problem has been studied when there is heterogeneity in one parameter (i.e., the distributional parameter), as well as when there is heterogeneity in two parameters (i.e., the distributional parameter and the mixing proportions). The sufficient conditions considered are based on p-larger order and reciprocally majorization order. Several examples have been provided to illustrate the established results.

Effective and Robust Detection of Adversarial Examples via Benford-Fourier Coefficients

Adversarial example has been well known as a serious threat to deep neural networks (DNNs). In this work, we study the detection of adversarial examples based on the assumption that the output and internal responses of one DNN model for both adversarial and benign examples follow the generalized Gaussian distribution (GGD) but with different parameters (i.e., shape factor, mean, and variance). GGD is a general distribution family that covers many popular distributions (e.g., Laplacian, Gaussian, or uniform). Therefore, it is more likely to approximate the intrinsic distributions of internal responses than any specific distribution. Besides, since the shape factor is more robust to different databases rather than the other two parameters, we propose to construct discriminative features via the shape factor for adversarial detection, employing the magnitude of Benford-Fourier (MBF) coefficients, which can be easily estimated using responses. Finally, a support vector machine is trained as an adversarial detector leveraging the MBF features. Extensive experiments in terms of image classification demonstrate that the proposed detector is much more effective and robust in detecting adversarial examples of different crafting methods and sources compared to state-of-the-art adversarial detection methods.

Family of Distributions Derived from Whittaker Function

In this paper, we introduce a general family of distributions based on Whittaker function. The properties of obtained distributions, moments, ordering, percentiles, and unimodality are studied. The distributions’ parameters are estimated using methods of moments and maximum likelihood. Furthermore, a generalization of Whittaker distribution that contains a wider class of distributions is developed. Validation of the obtained results is applied to real life data containing four data sets.

The U Family of Distributions: Properties and Applications

Abstract In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside-down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted…) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.

Does the Type of Records Affect the Estimates of the Parameters?

A general family of distributions, namely Kumaraswamy generalized family of (Kw-G) distribution, is considered for estimation of the unknown parameters and reliability function based on record data from Kw-G distribution. The maximum likelihood estimators (MLEs) are derived for unknown parameters and reliability function, along with its confidence intervals. A Bayesian study is carried out under symmetric and asymmetric loss functions in order to find the Bayes estimators for unknown parameters and reliability function. Future record values are predicted using Bayesian approach and non Bayesian approach, based on numerical examples and a monte carlo simulation.

A new flexible extension of the Lindley distribution with applications

A new general family of distributions based on the Lindley model was introduced. Some properties of the proposed model, including moments, quantiles, and order statistics are presented. In addition, some reliability measures of this model were investigated. The parameters were estimated using the moments and the maximum likelihood methods for complete and right-censored data. Then, the behavior of the maximum likelihood estimator was investigated in a simulation study. Finally, the model was applied to the analysis of two data sets to demonstrate its usefulness.

High-dimensional proportionality test of two covariance matrices and its application to gene expression data

With the development of modern science and technology, more and more high-dimensional data appear in the application fields. Since the high dimension can potentially increase the complexity of the covariance structure, comparing the covariance matrices among populations is strongly motivated in high-dimensional data analysis. In this article, we consider the proportionality test of two high-dimensional covariance matrices, where the data dimension is potentially much larger than the sample sizes, or even larger than the squares of the sample sizes. We devise a novel high-dimensional spatial rank test that has much-improved power than many existing popular tests, especially for the data generated from some heavy-tailed distributions. The asymptotic normality of the proposed test statistics is established under the family of elliptically symmetric distributions, which is a more general distribution family than the normal distribution family, including numerous commonly used heavy-tailed distributions. Extensive numerical experiments demonstrate the superiority of the proposed test in terms of both empirical size and power. Then, a real data analysis demonstrates the practicability of the proposed test for high-dimensional gene expression data.

Odd Burr III G-Negative Binomial Family with Application

Abstract In this article, we propose a general family of distributions called odd Burr III G-negative binomial. Several properties of the family are reported, including quantile function, mixture representation of its density, cumulative distribution functions, moments, incomplete moments, moment generating function, mean deviations, deviations, stochastic ordering, entropies, and parameter estimation via maximum likelihood (ML) method. A simulation study is carried out to check the asymptotic behavior of the ML estimates. The flexibility of the governing members of this family is shown by the use of data modeling such that the corresponding density functions may be symmetric, right-skewed, or left-skewed and their hazard rate functions can be increasing, decreasing, bathtub, upside-down bathtub, or constant.

Odd Kumaraswamy Fréchet Generated Family of Distributions with Applications

Odd Kumaraswamy Fréchet Generated Family of Distributions with Applications

General Results for General Transmuted-G Distributions

In this paper we have given some general results for the general transmuted families of distributions introduced by Rahman et al. (2018a,b). These results are helpful to obtain the results for any member of the general families of distributions and their special cases. We have given some examples for illustration.

An Expectation-Maximization Algorithm for the Exponential-Generalized Inverse Gaussian Regression Model with Varying Dispersion and Shape for Modelling the Aggregate Claim Amount

This article presents the Exponential–Generalized Inverse Gaussian regression model with varying dispersion and shape. The EGIG is a general distribution family which, under the adopted modelling framework, can provide the appropriate level of flexibility to fit moderate costs with high frequencies and heavy-tailed claim sizes, as they both represent significant proportions of the total loss in non-life insurance. The model’s implementation is illustrated by a real data application which involves fitting claim size data from a European motor insurer. The maximum likelihood estimation of the model parameters is achieved through a novel Expectation Maximization (EM)-type algorithm that is computationally tractable and is demonstrated to perform satisfactorily.

Ordering results between the largest claims arising from two general heterogeneous portfolios

This work is entirely devoted to compare the largest claims from two heterogeneous portfolios. It is assumed that the claim amounts in an insurance portfolio are nonnegative absolutely continuous random variables and belong to a general family of distributions. The largest claims have been compared based on various stochastic orderings. The established sufficient conditions are associated with the matrices and vectors of model parameters. Applications of the results are provided for the purpose of illustration.