Skewed Exponential Power Distribution Research Articles

Bayesian composite $$L^p$$-quantile regression

Abstract$$L^p$$ L p -quantiles are a class of generalized quantiles defined as minimizers of an asymmetric power function. They include both quantiles, $$p=1$$ p = 1 , and expectiles, $$p=2$$ p = 2 , as special cases. This paper studies composite $$L^p$$ L p -quantile regression, simultaneously extending single $$L^p$$ L p -quantile regression and composite quantile regression. A Bayesian approach is considered, where a novel parameterization of the skewed exponential power distribution is utilized. Further, a Laplace prior on the regression coefficients allows for variable selection. Through a Monte Carlo study and applications to empirical data, the proposed method is shown to outperform Bayesian composite quantile regression in most aspects.

Quantile regression based on the skewed exponential power distribution

Bayesian quantile regression generally relies on the asymmetric Laplace distribution (ALD) as the error distribution. We consider methods for Lp -quantile regression based on the skewed exponential power distribution (SEPD). Both Bayesian and frequentist estimation procedures are outlined and compared with previous work based on the SEPD. We find that our proposed methods greatly outperform a previous method in terms of quantile estimation. Further, compared with standard quantile regression, we find that our proposed methods generally perform better in terms of root mean square error (RMSE). Empirical evidence of the statistical properties of the proposed models is provided through a simulation study. Further, a real data application illustrates their performance.

Goodness-of-fit graphical assessment for a broad family of unimodal distributions

A Skewed Exponential Power Distribution is used to construct a bidimensional array of graphics in which kurtosis varies when moving horizontally and skewness when moving vertically. The Normal distribution lies in the center of the mosaic. Given a sample, the null hypotheses that it comes from each distribution in the mosaic are tested and the cell of each distribution is filled with a color that is a function of the obtained p-value. In this way, a dark area is obtained on the mosaic to indicate the distributions with which the sample is compatible.

Application of the skew exponential power distribution to ROC curves

The bi-Normal ROC model and corresponding metrics are commonly used in medical studies to evaluate the discriminatory ability of a biomarker. However, in practice, many clinical biomarkers tend to have skewed or other non-Normal distributions. And while the bi-Normal ROC model’s AUC tends to be unbiased in this setting, providing a reasonable measure of global performance, the corresponding decision thresholds tend to be biased. To correct this bias, we propose using an ROC model based on the skew exponential power (SEP) distribution, whose additional parameters can accommodate skewed, heavy tailed, or other non-Normal distributions. Additionally, the SEP distribution can be used to evaluate whether the bi-Normal model would be appropriate. The performance of these ROC models and the non-parametric approach are evaluated via a simulation study and applied to a real data set involving infections from Klebsiella pneumoniae. The SEP based ROC-model provides some efficiency gains with respect to estimation of the AUC and provides cut-points with improved classification rates. As such, in the presence non-Normal data, we suggest using the proposed SEP ROC model.

Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series

The goal of clustering is to identify common structures in a data set by forming groups of homogeneous objects. The observed characteristics of many economic time series motivated the development of classes of distributions that can accommodate properties, such as heavy tails and skewness. Thanks to its flexibility, the skewed exponential power distribution (also called skewed generalized error distribution) ensures a unified and general framework for clustering possibly skewed and heavy tailed time series. This paper develops a clustering procedure of model-based type, assuming that the time series are generated by the same underlying probability distribution but with different parameters. Moreover, we propose to optimally combine the estimated parameters to form the clusters with an entropy weighing k-means approach. The usefulness of the proposal is shown by means of application to financial time series, demonstrating also how the obtained clusters can be used to form portfolio of stocks.

Unified Bayesian conditional autoregressive risk measures using the skew exponential power distribution

Conditional Autoregressive Value-at-Risk and Conditional Autoregressive Expectile have become two popular approaches for direct measurement of market risk. Since their introduction several improvements both in the Bayesian and in the classical framework have been proposed to better account for asymmetry and local non-linearity. Here we propose a unified Bayesian Conditional Autoregressive Risk Measures approach by using the Skew Exponential Power distribution. Further, we extend the proposed models using a semiparametric P-Spline approximation answering for a flexible way to consider the presence of non-linearity. To make the statistical inference we adapt the MCMC algorithm proposed in Bernardi et al. (2018) to our case. The effectiveness of the whole approach is demonstrated using real data on daily return of five stock market indices.

Bayesian Analysis of the Epsilon Skew Exponential Power Distribution

Bayesian Analysis of the Epsilon Skew Exponential Power Distribution

Loss-based approach to two-piece location-scale distributions with applications to dependent data

Two-piece location-scale models are used for modeling data presenting departures from symmetry. In this paper, we propose an objective Bayesian methodology for the tail parameter of two particular distributions of the above family: the skewed exponential power distribution and the skewed generalised logistic distribution. We apply the proposed objective approach to time series models and linear regression models where the error terms follow the distributions object of study. The performance of the proposed approach is illustrated through simulation experiments and real data analysis. The methodology yields improvements in density forecasts, as shown by the analysis we carry out on the electricity prices in Nordpool markets.

Robust linear mixed models for Small Area Estimation

Hierarchical models are popular in many applied statistics fields including Small Area Estimation. One well known model employed in this particular field is the Fay–Herriot model, in which unobservable parameters are assumed to be Gaussian. In Hierarchical models assumptions about unobservable quantities are difficult to check. For a special case of the Fay–Herriot model, Sinharay and Stern [2003. Posterior predictive model checking in Hierarchical models. J. Statist. Plann. Inference 111, 209–221] showed that violations of the assumptions about the random effects are difficult to detect using posterior predictive checks. In this present paper we consider two extensions of the Fay–Herriot model in which the random effects are assumed to be distributed according to either an exponential power (EP) distribution or a skewed EP distribution. We aim to explore the robustness of the Fay–Herriot model for the estimation of individual area means as well as the empirical distribution function of their ‘ensemble’. Our findings, which are based on a simulation experiment, are largely consistent with those of Sinharay and Stern as far as the efficient estimation of individual small area parameters is concerned. However, when estimating the empirical distribution function of the ‘ensemble’ of small area parameters, results are more sensitive to the failure of distributional assumptions.

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.

Inferential Aspects of the Skew Exponential Power Distribution

When the analysis of real data indicates that normality assumptions are untenable, more flexible models that cope with the most prevalent deviations from normality can be adopted. In this context, the skew exponential power (SEP) distribution warrants special attention, because it encompasses distributions having both heavy tails and skewness, it allows likelihood inference, and it includes the normal model as a special case. This article concerns likelihood inference about the parameters of the SEP family. In particular, the information matrix of the maximum likelihood estimators (MLEs) is obtained and finite-sample properties of the estimators are investigated numerically. Special attention is given to the properties of the MLEs and likelihood ratio statistics when the data are drawn from a normal distribution, because this case is relevant for using the SEP distribution to test for normality. Application of the SEP distribution in robust estimation problems is considered for both independent and dependent data. Under moderate deviations from normality, estimators obtained under the SEP distribution are shown to outperform the normal-based estimators and to compete with robust estimators; furthermore, the SEP distribution offers the benefits of having a specified distribution, such as interpretable location and scale parameters under nonnormality.