Exponential Power Distribution Research Articles

A new heavy tailed distribution with actuarial measures

Actuaries are constantly on the lookout for heavy-tailed (HT) distributions in order to model data important to business and actuarial risk problems. In this article, we describe a novel type of heavy-tailed distribution that may be used to model data in the financial disciplines. Our proposed model, identified as a Kavya-Manoharan power Lomax (KMPLo) distribution, is thoroughly studied. Some fundamental characterizations, such as the quantile function and moments, have been developed. The maximum likelihood (ML) and Bayesian (B) analytical techniques are used to acquire estimates of the unknown parameters of the new model. A detailed simulation study is undertaken to evaluate the performance of the ML and B estimators. The KMPLo model's utility is demonstrated by an application to a HT insurance loss data set. The experimental results reveal that the suggested model is more adaptable and affordable than the other seven competing models including (i) the four-parameter model is the power Lomax (PLo) model; (ii) the four-parameter models odd log-logistic modified Weibull (OLLMW), Kumaraswamy Weibull (KW), generalized modified Weibull (GMW), extended odd Weibull Lomax (EOWL), Weibull-Lomax (WL), Marshall–Olkin power Lomax (MOPLo), Marshall–Olkin alpha power Lomax (MOAPLo) and exponentiated generalized alpha power exponential (EGAPEx) distributions (iii) the five-parameter distributions; , Kumaraswamy generalized power Lomax (KGPLo) and Marshall-Olkin alpha power exponentiated Weibull (MOAPEW) distribution. In addition, certain essential actuarial metrics, such like value at risk and conditional value at risk, are determined.

Elliptical Capital Asset Pricing Models: Formulation, Diagnostics, Case Study with Chilean Data, and Economic Rationale

The capital asset pricing model (CAPM) is often based on the Gaussianity or normality assumption. However, such an assumption is frequently violated in practical situations. In this paper, we introduce the symmetric CAPM considering distributions with lighter or heavier tails than the normal distribution. These distributions are symmetric and belong to the family of elliptical distributions. We pay special attention to the family members related to the normal, power-exponential, and Student-t cases, with the power-exponential distribution being particularly considered, as it has not been explored widely. Based on these cases, the expectation-maximization algorithm can be used to facilitate the estimation of model parameters utilizing the maximum likelihood method. In addition, we derive the leverage and local influence methods to carry out diagnostics in the symmetric CAPM. We conduct a detailed case study to apply the obtained results estimating the systematic risk of the financial assets of a Chilean company with real data. We employ the Akaike information criterion to conclude that the studied models provide better results than the CAPM under Gaussianity.

Model-Based Clustering and Classification Using Mixtures of Multivariate Skewed Power Exponential Distributions

Families of mixtures of multivariate power exponential (MPE) distributions have already been introduced and shown to be competitive for cluster analysis in comparison to other mixtures of elliptical distributions, including mixtures of Gaussian distributions. A family of mixtures of multivariate skewed power exponential distributions is proposed that combines the flexibility of the MPE distribution with the ability to model skewness. These mixtures are more robust to variations from normality and can account for skewness, varying tail weight, and peakedness of data. A generalized expectation-maximization approach, which combines minorization-maximization and optimization based on accelerated line search algorithms on the Stiefel manifold, is used for parameter estimation. These mixtures are implemented both in the unsupervised and semi-supervised classification frameworks. Both simulated and real data are used for illustration and comparison to other mixture families.

A Mixture Autoregressive Model Based on an Asymmetric Exponential Power Distribution

In nonlinear time series analysis, the mixture autoregressive model (MAR) is an effective statistical tool to capture the multimodality of data. However, the traditional methods usually need to assume that the error follows a specific distribution that is not adaptive to the dataset. This paper proposes a mixture autoregressive model via an asymmetric exponential power distribution, which includes normal distribution, skew-normal distribution, generalized error distribution, Laplace distribution, asymmetric Laplace distribution, and uniform distribution as special cases. Therefore, the proposed method can be seen as a generalization of some existing model, which can adapt to unknown error structures to improve prediction accuracy, even in the case of fat tail and asymmetry. In addition, an expectation-maximization algorithm is applied to implement the proposed optimization problem. The finite sample performance of the proposed approach is illustrated via some numerical simulations. Finally, we apply the proposed methodology to analyze the daily return series of the Hong Kong Hang Seng Index. The results indicate that the proposed method is more robust and adaptive to the error distributions than other existing methods.

Parameters and Reliability Estimation of Alpha Power Exponential Distribution under Type-II Progressive Hybrid Censoring with Applications of Engineering and Management Fields

Parameters and Reliability Estimation of Alpha Power Exponential Distribution under Type-II Progressive Hybrid Censoring with Applications of Engineering and Management Fields

MoEP-AE: Autoencoding Mixtures of Exponential Power Distributions for Open-Set Recognition

Open-set recognition aims to identify unknown classes while maintaining classification performance on known classes and has attracted increasing attention in the pattern recognition field. However, how to learn effective feature representations whose distributions are usually complex for classifying both known-class and unknown-class samples when only the known-class samples are available for training is an ongoing issue in open-set recognition. In contrast to methods implementing a single Gaussian, a mixture of Gaussians (MoG), or multiple MoGs, we propose a novel autoencoder that learns feature representations by modeling them as mixtures of exponential power distributions (MoEPs) in latent spaces called MoEP-AE. The proposed autoencoder considers that many real-world distributions are sub-Gaussian or super-Gaussian and can thus be represented by MoEPs rather than a single Gaussian or an MoG or multiple MoGs. We design a differentiable sampler that can sample from an MoEP to guarantee that the proposed autoencoder is trained effectively. Furthermore, we propose an MoEP-AE-based method for open-set recognition by introducing a discrimination strategy, where the MoEP-AE is used to model the distributions of the features extracted from the input known-class samples by minimizing a designed loss function at the training stage, called MoEP-AE-OSR. Extensive experimental results in both standard-dataset and cross-dataset settings demonstrate that the MoEP-AE-OSR method outperforms 14 existing open-set recognition methods in most cases in both open-set recognition and closed-set recognition tasks.

Does the asymmetric exponential power distribution improve systemic risk measurement?

Does the asymmetric exponential power distribution improve systemic risk measurement?

New Lifetime Distribution for Modeling Data on the Unit Interval: Properties, Applications and Quantile Regression

Probability distributions are very useful in modeling lifetime datasets. However, no specific distribution is suitable for all kinds of datasets. In this study, the bounded truncated Cauchy power exponential distribution is proposed for modeling datasets on the unit interval. The probability density function exhibits desirable shapes, such as left-skewed, right-skewed, reversed J, and bathtub shapes, whereas the hazard rate function displays J and bathtub shapes. For the purpose of modeling dependence between measures in a dataset, a bivariate extension of the proposed distribution is developed. The bivariate probability density function displays monotonic and non-monotonic shapes, making it suitable for modeling complex bivariate relations. Subsequently, the applications of the distribution are illustrated using COVID-19 data. The results revealed that the new distribution provides a better fit to the datasets compared to other existing distributions. Finally, a new quantile regression model is developed and its application demonstrated. The generated quantile regression model offers a decent fit to the data, according to the residual analysis.

Goodness-of-fit tests for Laplace, Gaussian and exponential power distributions based on λ-th power skewness and kurtosis

Temperature data, like many other measurements in quantitative fields, are usually modelled using a normal distribution. However, some distributions can offer a better fit while avoiding underestimation of tail event probabilities. To this point, we extend Pearson's notions of skewness and kurtosis to build a powerful family of goodness-of-fit tests based on Rao's score for the exponential power distribution , including tests for normality and Laplacity when λ is set to 1 or 2. We find the asymptotic distribution of our test statistic, which is the sum of the squares of two Z-scores, under the null and under local alternatives. We also develop an innovative regression strategy to obtain Z-scores that are nearly independent and distributed as standard Gaussians, resulting in a distribution valid for any sample size (up to very high precision for ). The case leads to a powerful test of fit for the Laplace() distribution, whose empirical power is superior to all 39 competitors in the literature, over a wide range of 400 alternatives. Theoretical proofs in this case are particularly challenging and substantial. We applied our tests to three temperature datasets. The new tests are implemented in the R package PoweR.

Topp–Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data

In this article, a four parameter lifetime model called the Topp–Leone modified Weibull distribution is proposed. The suggested distribution can be considered as an alternative to Kumaraswamy Weibull, generalized modified Weibull, extend odd Weibull Lomax, Weibull-Lomax, Marshall-Olkin alpha power extended Weibull and exponentiated generalized alpha power exponential distributions, etc. The suggested model includes the Topp-Leone Weibull, Topp-Leone Linear failure rate, Topp-Leone exponential and Topp-Leone Rayleigh distributions as a special case. Several characteristics of the new suggested model including quantile function, moments, moment generating function, central moments, mean, variance, coefficient of skewness, coefficient of kurtosis, incomplete moments, the mean residual life and the mean inactive time are derived. The probability density function of the Topp–Leone modified Weibull distribution can be right skewed and uni-modal shaped but, the hazard rate function may be decreasing, increasing, J-shaped, U-shaped and bathtub on its parameters. Three different methods of estimation as; maximum likelihood, maximum product spacing and Bayesian methods are used to estimate the model parameters. For illustrative reasons, applications of the Topp–Leone modified Weibull model to four real data sets related to medical and engineering sciences are provided and contrasted with the fit reached by several other well-known distributions.

Statistical Analysis of Alpha Power Exponential Parameters Using Progressive First-Failure Censoring with Applications

This paper is an endeavor to investigate some estimation problems of the unknown parameters and some reliability measures of the alpha power exponential distribution in the presence of progressive first-failure censored data. In this regard, the classical and Bayesian approaches are considered to acquire the point and interval estimates of the different quantities. The maximum likelihood approach is proposed to obtain the estimates of the unknown parameters, reliability, and hazard rate functions. The approximate confidence intervals are also considered. The Bayes estimates are obtained by considering both symmetric and asymmetric loss functions. The Bayes estimates and the associated highest posterior density credible intervals are given by applying the Monte Carlo Markov Chain technique. Due to the complexity of the given estimators which cannot be compared theoretically, a simulation study is implemented to compare the performance of the different procedures. In addition, diverse optimality criteria are employed to pick the best progressive censoring plans. Two engineering applications are considered to illustrate the applicability of the offered estimators. The numerical outcomes showed that the Bayes estimates based on symmetric or asymmetric loss functions perform better than other estimates in terms of minimum root mean square errors and interval lengths.

Parameter estimation procedures for log exponential-power distribution with real data applications

In this study, some estimation techniques were investigated to estimate of parameters of the log exponential-power distribution. Maximum likelihood, quantile, least squares, weighted least squares, Anderson-Darling, and Cramer-von Mises estimation methods are studied in detail. The efficiency of these estimators was validated through Monte Carlo simulation experiments. To assess the performance of these estimators, Monte Carlo simulation studies were conducted out. Also, four real data applications are performed and Kolmogorov-Smirnov statistic results for all estimators are presented.

Information Approach for Change Point Analysis of EGGAPE Distribution and Application to COVID-19 Data

The exponentiated generalized Gull alpha power exponential distribution is an extension of the exponential distribution that can model data characterized by various shapes of the hazard function. However, change point problem has not been studied for this distribution. In this study, the change point detection of the parameters of the exponentiated generalized Gull alpha power exponential distribution is studied using the modified information criterion. In addition, the binary segmentation procedure is used to identify multiple change point locations. The assumption is that all the parameters of the EGGAPE distributions are considered changeable. Simulation study is conducted to illustrate the power of the modified information criterion in detecting change point in the parameters with different sample sizes. Three applications related to COVID-19 data are used to demonstrate the applicability of the MIC in detecting change point in real life scenario.

Classical and Bayesian Estimation in Exponential Power Distribution under Type-I Progressive Hybrid Censoring with Binomial Removals

This article deals with the classical and Bayesian estimation in exponential power distribution based on Type-I progressive hybrid censoring with binomial removals at each stage. Based on the considered censoring scheme, the maximum likelihood estimates and their coverage probabilities are computed by the Monte Carlo simulation technique. MCMC technique is used to obtain the Bayes estimates under the informative priors. The performance of both the approaches is evaluated in terms of their absolute bias and mean square error (MSE) as well as the width of the confidence interval. Applicability of the suggested approach is illustrated by analysis of a real-life dataset.

Inferences of a Mixture Bivariate Alpha Power Exponential Model with Engineering Application

The univariate alpha power exponential (APE) distribution has several appealing characteristics. It behaves similarly to Weibull, Gamma, and generalized exponential distributions with two parameters. In this paper, we consider different bivariate mixture models starting with two independent univariate APE models, and, in the latter case, starting from two dependent APE models. Several useful structural properties of such a mixture model (under the assumption of two independent APE distribution) are discussed. Bivariate APE (BAPE), in short, modelled under the dependent set up are also discussed in the context of a copula-based construction. Inferential aspects under the classical and under the Bayesian paradigm are considered to estimate the model parameters, and a simulation study is conducted for this purpose. For illustrative purposes, a well-known motor data is re-analyzed to exhibit the flexibility of the proposed bivariate mixture model.

A Generalized Form of Power Transformation on Exponential Family of Distribution with Properties and Application

In this paper, we proposed a new generalized family of distribution namely new alpha power Exponential (NAPE) distribution based on the new alpha power transformation (NAPT) method by Elbatal et al. (2019). Various statistical properties of the proposed distribution are obtained including moment, incomplete moment, conditional moment, probability weighted moments (PWMs), quantile function, residual and reversed residual lifetime function, stress-strength parameter, entropy and order statistics. The percentage point of NAPE distribution for some specific values of the parameters is also obtained. The method of maximum likelihood estimation (MLE) has been used for estimating the parameters of NAPE distribution. A simulation study has been performed to evaluate and execute the behavior of the estimated parameters for mean square errors (MSEs) and bias. Finally, the efficiency and flexibility of the new proposed model are illustrated by analyzing three real-life data sets.

Estimation of Reliability Indices for Alpha Power Exponential Distribution Based on Progressively Censored Competing Risks Data

In reliability analysis and life testing studies, the experimenter is frequently interested in studying a specific risk factor in the presence of other factors. In this paper, the estimation of the unknown parameters, reliability and hazard functions of alpha power exponential distribution is considered based on progressively Type-II censored competing risks data. We assume that the latent cause of failures has independent alpha power exponential distributions with different scale and shape parameters. The maximum likelihood method is considered to estimate the model parameters as well as the reliability and hazard rate functions. The approximate and two parametric bootstrap confidence intervals of the different estimators are constructed. Moreover, the Bayesian estimation method of the unknown parameters, reliability and hazard rate functions are obtained based on the squared error loss function using independent gamma priors. To get the Bayesian estimates as well as the highest posterior credible intervals, the Markov Chain Monte Carlo procedure is implemented. A comprehensive simulation experiment is conducted to compare the performance of the proposed procedures. Finally, a real dataset for the relapse of multiple myeloma with transplant-related mortality is analyzed.

Optimal Plan of Multi-Stress–Strength Reliability Bayesian and Non-Bayesian Methods for the Alpha Power Exponential Model Using Progressive First Failure

It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little attention to. The goal of this paper is to develop inference for multi-reliability using unit alpha power exponential distributions for stress–strength variables based on the progressive first failure. As a result, the problem of estimating the stress–strength function R, where X, Y, and Z come from three separate alpha power exponential distributions, is addressed in this paper. The conventional methods, such as maximum likelihood for point estimation, Bayesian and asymptotic confidence, boot-p, and boot-t methods for interval estimation, are also examined. Various confidence intervals have been obtained. Monte Carlo simulations and real-world application examples are used to evaluate and compare the performance of the various proposed estimators.

Upper bound for variance of finite mixtures of power exponential distributions

Abstract Both variance and entropy are commonly used measures for uncertainty. There exists many cases where variance is infinite and entropy is finite. In this note, we derive an upper bound illustrating the relationship between variance and entropy of random variables having a special class of distributions. We also derive an upper bound for kth absolute central moment proportional to entropy power for a special class of distributions.

The Exponentiated Generalized Alpha Power Family of Distribution: Properties and Applications

In this paper, we introduce the exponentiated generalized alpha power family of distributions to extend the several other distributions. We used the new family to discuss the exponentiated generalized alpha power exponential (EGAPEx) distribution. Some statistical properties of the EGAPEx distribution are obtained. The model parameters are obtained by the maximum likelihood estimation (MLE), maximum product spacing (MPS) and Bayesian estimation methods. A Monte Carlo Simulation is performed to compare between different methods. We illustrate the performance of the proposed new family of distributions by means of two real data sets and the data sets show the new family of distributions is more appropriate as compared to the exponentiated generalized exponential, alpha power generalized exponential, alpha power exponential, generalized exponential and exponential distributions.