Family Of Heavy-tailed Distributions Research Articles

Feller-Pareto and Related Distributions: Numerical Implementation and Actuarial Applications

Actuaries model insurance claim amounts using heavy tailed probability distributions. They routinely need to evaluate quantities related to these distributions such as quantiles in the far right tail, moments or limited moments. Furthermore, actuaries often resort to simulation to solve otherwise untractable risk evaluation problems. The paper discusses our implementation of support functions for the Feller-Pareto distribution for the R package actuar. The Feller-Pareto defines a large family of heavy tailed distributions encompassing the transformed beta family and many variants of the Pareto distribution.

Logistic Quantile Regression for Bounded Outcomes Using a Family of Heavy-Tailed Distributions

Mean regression model could be inadequate if the probability distribution of the observed responses is not symmetric. Under such situation, the quantile regression turns to be a more robust alternative for accommodating outliers and misspecification of the error distribution, since it characterizes the entire conditional distribution of the outcome variable. This paper proposes a robust logistic quantile regression model by using a logit link function along the EM-based algorithm for maximum likelihood estimation of the p th quantile regression parameters in Galarza (Stat 6, 1, 2017). The aforementioned quantile regression (QR) model is built on a generalized class of skewed distributions which consists of skewed versions of normal, Student’s t, Laplace, contaminated normal, slash, among other heavy-tailed distributions. We evaluate the performance of our proposal to accommodate bounded responses by investigating a synthetic dataset where we consider a full model including categorical and continuous covariates as well as several of its sub-models. For the full model, we compare our proposal with a non-parametric alternative from the so-called quantreg R package. The algorithm is implemented in the R package lqr, providing full estimation and inference for the parameters, automatic selection of best model, as well as simulation of envelope plots which are useful for assessing the goodness-of-fit.

The Arcsine Exponentiated-X Family: Validation and Insurance Application

In this paper, we propose a family of heavy tailed distributions, by incorporating a trigonometric function called the arcsine exponentiated-X family of distributions. Based on the proposed approach, a three-parameter extension of the Weibull distribution called the arcsine exponentiated-Weibull (ASE-W) distribution is studied in detail. Maximum likelihood is used to estimate the model parameters, and its performance is evaluated by two simulation studies. Actuarial measures including Value at Risk and Tail Value at Risk are derived for the ASE-W distribution. Furthermore, a numerical study of these measures is conducted proving that the proposed ASE-W distribution has a heavier tail than the baseline Weibull distribution. These actuarial measures are also estimated from insurance claims real data for the ASE-W and other competing distributions. The usefulness and flexibility of the proposed model is proved by analyzing a real-life heavy tailed insurance claims data. We construct a modified chi-squared goodness-of-fit test based on the Nikulin–Rao–Robson statistic to verify the validity of the proposed ASE-W model. The modified test shows that the ASE-W model can be used as a good candidate for analyzing heavy tailed insurance claims data.

A new family of heavy tailed distributions with an application to the heavy tailed insurance loss data

Heavy tailed distributions play very significant role in the study of actuarial and financial risk management data but the probability distributions proposed to model such data are scanty. Actuaries often search for new and appropriate statistical models to address data related to financial risk problems. In this work, we propose a new family of heavy tailed distributions. Some basic properties of this new family of heavy tailed distributions are obtained. A special sub-model of the proposed family, called a new heavy tailed Weibull model is considered in detail. The maximum likelihood estimators of the model parameters are obtained. A Monte Carlo simulation study is carried out to evaluate the performance of these estimators. Furthermore, some actuarial measures such as value at risk and tail value at risk are calculated. A simulation study based on these actuarial measures is done. Finally, an application of the proposed model to a heavy tailed insurance loss data set is presented.

Modelling Insurance Losses with a New Family of Heavy-Tailed Distributions

The actuaries always look for heavy-tailed distributions to model data relevant to business and actuarial risk issues. In this article, we introduce a new class of heavy-tailed distributions useful for modeling data in financial sciences. A specific sub-model form of our suggested family, named as a new extended heavy-tailed Weibull distribution is examined in detail. Some basic characterizations, including quantile function and raw moments have been derived. The estimates of the unknown parameters of the new model are obtained via the maximum likelihood estimation method. To judge the performance of the maximum likelihood estimators, a simulation analysis is performed in detail. Furthermore, some important actuarial measures such as value at risk and tail value at risk are also computed. A simulation study based on these actuarial measures is conducted to exhibit empirically that the proposed model is heavy-tailed. The usefulness of the proposed family is illustrated by means of an application to a heavy-tailed insurance loss data set. The practical application shows that the proposed model is more flexible and efficient than the other six competing models including (i) the two-parameter models Weibull, Lomax and Burr-XII distributions (ii) the three-parameter distributions Marshall-Olkin Weibull and exponentiated Weibull distributions, and (iii) a well-known four-parameter Kumaraswamy Weibull distribution.

DEA model considering outputs with stochastic noise and a heavy-tailed (stable) distribution

Data envelopment analysis (DEA) is a nonparametric data-based approach that is used to evaluate the performance of a set of homogeneous units; hence, inputs and outputs play a crucial role in this evaluation. In many technologies, the inputs and outputs are so volatile and complex that even other stochastic events can have an influence in the measurement accuracy. A suitable estimation method can be selected according to the type of stochastic model used. If the model has stochastic noises, then in the first step, the noise must be identified by its stochastic behaviour as well as a probability distribution. In DEA, it is important to transform the probabilistic form of a model to its equivalent deterministic model. The chance-constrained programming input (output) oriented model with a normal distribution assumption is most often applied in stochastic cases. However, this study improves the assumption by using a family of heavy-tailed distributions, which is called Stable Distributions. It is assumed that the inputs are deterministic and that the outputs include stochastic noise with a stable distribution. The interesting properties of the selected distribution family help us to solve the probability constraints. A numerical example is provided to illustratively validate the proposed approach.

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACTPareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD(IV) distribution. Various properties of the GPD(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD(IV) distribution.

Alpha-Stable Matrix Factorization

Matrix factorization (MF) models have been widely used in data analysis. Even though they have been shown to be useful in many applications, classical MF models often fall short when the observed data are impulsive and contain outliers. In this study, we present $\alpha $ MF, a MF model with $\alpha $ -stable observations. Stable distributions are a family of heavy-tailed distributions that is particularly suited for such impulsive data. We develop a Markov Chain Monte Carlo method, namely a Gibbs sampler, for making inference in the model. We evaluate our model on both synthetic and real audio applications. Our experiments on speech enhancement show that $\alpha $ MF yields superior performance to a popular audio processing model in terms of objective measures. Furthermore, $\alpha $ MF provides a theoretically sound justification for recent empirical results obtained in audio processing.

A Robust Estimator of R = P(X > Y ) of Heavy-tailed Distributions and its Sampling Distributions

Heavy-tailed distributions have wide ap- plications in life-time contexts, especially in reliability and risk modeling. So we consider the estimation problem of reliability, R = P (X > Y ) for small samples, when X and Y are two independent but not identically distributed random variables belonging to the family of heavy-tailed distributions, using a robust estimator, namely the harmonic moment estimator. Extensive sim- ulation studies are carried out to study the performance of this estimator. The relative efficiency of the estima- tor with the well known Hill estimator is studied. We obtain the sampling distribution of the parameters of the distribution as well as that of estimator of R which will help us to study the properties of the estimators. Also we find out the asymptotic confidence intervals of R and its performance is studied with respect to average width and the coverage probability, through simulations. especially in reliability and risk modeling. We consider the estimation problem of reliability R = P (X > Y ) for small and large samples, when X and Y are two inde- pendent but not identically distributed random variables belonging to the family of heavy-tailed distributions.The heavy-tailed distributions like the student's t and Pareto families have been used to model data with high kurto- sis see, (4). The simplest heavy tailed distribution is the Pareto distribution. It is hyperbolic over its entire range. The Pareto distribution is a power law probability dis- tribution that coincides with social, scientific, geophys- ical, actuarial and many other types of observable phe- nomena. The univariate Pareto distribution is a simple model for non-negative data with a power law probabil- ity tail, at least approximately for large values of X. It is a useful model in the analysis of income data, reliabil- ity studies, risk modeling and business failure data (14). Reference (1) gave an extensive historical survey of its uses in the context of income distribution. References (6), (16) and (19) discussed the application of the Pareto disrribution in various fields. Moreover Pareto Type I distribution has a position of importance in the field of life testing because this distribution can be considered as a distribution of failure time of a component as fol- lows.

A Marshall–Olkin family of heavy-tailed distributions which includes the lognormal one

A new class of heavy-tailed distribution functions,, containing the lognormal distribution as a particular case is introduced. The class thus obtained depends on a set of three parameters, incorporating an additional distribution to the classical lognormal one. This new class of heavy-tailed distribution is presented as an alternative to other useful heavy-tailed distributions, such as the lognormal, Weibull, and Pareto distributions. The density and distribution functions of this new class are given by a closed expression which allows us to easily compute probabilities, quantiles, moments, and related measurements. Finally, some applications are shown as examples.

Improved Gabor Filter for Extracting Texture Edge Features in Ultrasound Kidney Images

The diagnosis of urinary tract calculi begins with a focused history of calculi, evolution of its symptoms and the duration. Computer assisted approaches for analyzing the images have increased since the manual interpretation of the image is a time consuming process and susceptible to human errors. The proposed system develops a multi-scale wavelet based Bayesian speckle suppression method for ultrasound kidney images. The logarithmic transform of the original image is analyzed into the multi-scale wavelet domain. The subband decompositions of ultrasound images have significantly non-Gaussian statistics that are best described by families of heavy-tailed distributions. Bayesian estimators are designed to exploits these statistics. Ultrasound (US) is increasingly considered as a viable alternative imaging modality in computer-assisted Kidney segmentation and disease diagnosis applications. Automatic Kidney segmentation from US images, however, remains a challenge due to speckle noise and various other artifacts inherent to US. This paper, design intensity invariant local image phase features, obtained using improved Gabor filter banks, for extracting edge texture features that occur at core and intermediate layer interfaces. The proposed model does the extension of phase symmetry features to modified gabor mode and their use in automatic extraction of kidney edge texture features from US normal and diseased patient images. The system functionality is proved qualitatively and quantitatively through experimentation for synthetic and real data sets. The localization feature value threshold is evaluated with the training samples of US images. The speckle noise error ratio with respect to the standard US image are compared and experimented.

Robust Sequential Learning Algorithms for Linear Observation Models

This paper presents a study of sequential parameter estimation based on a linear non-Gaussian observation model. To develop robust algorithms, we consider a family of heavy-tailed distributions that can be expressed as the scale mixture of Gaussian and extend the development to include some robust penalty functions. We treat the problem as a Bayesian learning problem and develop an iterative algorithm by using the Laplace approximation for the posterior and the minorization-maximization (MM) algorithm as an optimization tool. We then study a one-step implementation of the iterative algorithm. This leads to a family of generalized robust RLS-type of algorithms which include several well-known algorithms as special cases. Using a further simplification that the covariance is fixed, leads to a family of generalized robust LMS-type of algorithms. Through mathematical analysis and simulations, we demonstrate the robustness of these algorithms