Skewed Exponential Power Research Articles

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

Abstract$$L^p$$Lp-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$$L^p$$Lp-quantile regression, simultaneously extending single$$L^p$$Lp-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.

FitFun: A modelling framework for successfully capturing the functional form and noise of observed traffic flow–density–speed relationships

Measurements of the average properties of vehicular traffic are inherently noisy. The distributions of flow and speed measurements at any particular density are non-Gaussian with density-dependent variance, skewness, and kurtosis. Previous studies have failed to properly account for these complicated noise properties. In remediation, we present FitFun, a general framework for modelling any observed flow–density–speed relationship. Models specified within FitFun incorporate components for both the functional form and the noise. We define three flexible noise model components and we fit 200 different models to a high-quality sample of 10,150 observed urban flow-occupancy relationships. We compare the fits using information criteria and assess fit quality through analysis of the residuals. We find that the non-parametric Sun model for the functional form component combined with a Skew Exponential Power Type III noise component significantly outperforms all of the other models. Interestingly, we find that the city, country, road topology, and detector location have virtually no impact on model performance and fit quality, which is very convenient for model selection. The only factor of relevance from those that we studied is the effective occupancy coverage of the data. We conclude that certain models specified judiciously within FitFun can successfully capture the functional form and noise of observed flow–density–speed relationships without the need to discard data taken during non-stationary conditions. This is particularly advantageous for urban data where stationary traffic conditions are rarely observed.

Using Skewed Exponential Power Mixture for VaR and CVaR Forecasts to Comply with Market Risk Regulation

Using Skewed Exponential Power Mixture for VaR and CVaR Forecasts to Comply with Market Risk Regulation

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.

Forecasting VaR and CVaR Based on a Skewed Exponential Power Mixture, in Compliance With the New Market Risk Regulation

Forecasting VaR and CVaR Based on a Skewed Exponential Power Mixture, in Compliance With the New Market Risk Regulation

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

Mosaic normality test

A procedure is proposed here for jointly visualizing the compatibility of a sample with a family of Skewed Exponential Power Distributions, of which the distributions known as Normal, Exponential, Laplace and Uniform are particular cases. The procedure involves constructing a mosaic that contains these distributions in such a way that the asymmetry varies from left to right and the kurtosis varies from top to bottom. The null hypothesis that the sample belongs to each of the mosaic distributions is tested, with the corresponding box for each distribution being shaded in a gray scale according to the p-value obtained. The location and shape of the shaded area that appears on the mosaic facilitates not only identifying which distributions are compatible with the sample, but also assessing the power of the test performed. All the parameters that define the distribution are considered to be known. The problem remains open for expanding the procedure to cases in which these parameters are not known but must be estimated from the sample. As additional material, a code written in R that carries out the proposed test is included.

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.

Bayesian quantile regression using the skew exponential power distribution

Traditional Bayesian quantile regression relies on the Asymmetric Laplace (AL) distribution due primarily to its satisfactory empirical and theoretical performances. However, the AL displays medium tails and it is not suitable for data characterized by strong deviations from the Gaussian hypothesis. An extension of the AL Bayesian quantile regression framework is proposed to account for fat tails using the Skew Exponential Power (SEP) distribution. Linear and Additive Models (AM) with penalized splines are considered to show the flexibility of the SEP in the Bayesian quantile regression context. Lasso priors are used in both cases to account for the problem of shrinking parameters when the parameters space becomes wide while Bayesian inference is implemented using a new adaptive Metropolis within Gibbs algorithm. Empirical evidence of the statistical properties of the proposed models is provided through several examples based on both simulated and real datasets.

Joint analysis of input and parametric uncertainties in watershed water quality modeling: A formal Bayesian approach

Abstract Significant Input uncertainty is a major source of error in watershed water quality (WWQ) modeling. It remains challenging to address the input uncertainty in a rigorous Bayesian framework. This study develops the Bayesian Analysis of Input and Parametric Uncertainties (BAIPU), an approach for the joint analysis of input and parametric uncertainties through a tight coupling of Markov Chain Monte Carlo (MCMC) analysis and Bayesian Model Averaging (BMA). The formal likelihood function for this approach is derived considering a lag-1 autocorrelated, heteroscedastic, and Skew Exponential Power (SEP) distributed error model. A series of numerical experiments were performed based on a synthetic nitrate pollution case and on a real study case in the Newport Bay Watershed, California. The Soil and Water Assessment Tool (SWAT) and Differential Evolution Adaptive Metropolis (DREAM(ZS)) were used as the representative WWQ model and MCMC algorithm, respectively. The major findings include the following: (1) the BAIPU can be implemented and used to appropriately identify the uncertain parameters and characterize the predictive uncertainty; (2) the compensation effect between the input and parametric uncertainties can seriously mislead the modeling based management decisions, if the input uncertainty is not explicitly accounted for; (3) the BAIPU accounts for the interaction between the input and parametric uncertainties and therefore provides more accurate calibration and uncertainty results than a sequential analysis of the uncertainties; and (4) the BAIPU quantifies the credibility of different input assumptions on a statistical basis and can be implemented as an effective inverse modeling approach to the joint inference of parameters and inputs.

Effect of heteroscedasticity treatment in residual error models on model calibration and prediction uncertainty estimation

The heteroscedasticity treatment in residual error models directly impacts the model calibration and prediction uncertainty estimation. This study compares three methods to deal with the heteroscedasticity, including the explicit linear modeling (LM) method and nonlinear modeling (NL) method using hyperbolic tangent function, as well as the implicit Box-Cox transformation (BC). Then a combined approach (CA) combining the advantages of both LM and BC methods has been proposed. In conjunction with the first order autoregressive model and the skew exponential power (SEP) distribution, four residual error models are generated, namely LM-SEP, NL-SEP, BC-SEP and CA-SEP, and their corresponding likelihood functions are applied to the Variable Infiltration Capacity (VIC) hydrologic model over the Huaihe River basin, China. Results show that the LM-SEP yields the poorest streamflow predictions with the widest uncertainty band and unrealistic negative flows. The NL and BC methods can better deal with the heteroscedasticity and hence their corresponding predictive performances are improved, yet the negative flows cannot be avoided. The CA-SEP produces the most accurate predictions with the highest reliability and effectively avoids the negative flows, because the CA approach is capable of addressing the complicated heteroscedasticity over the study basin.

Skew exponential power stochastic volatility model for analysis of skewness, non-normal tails, quantiles and expectiles

This paper proposes a unified framework to analyse the skewness, tail heaviness, quantiles and expectiles of the return distribution based on a stochastic volatility model using a new parametrisati...

Measuring Inflation Expectations Using Interval‐Coded Data*

AbstractTo quantify qualitative survey data, the Carlson–Parkin method assumes normality, a time‐invariant symmetric indifference interval, and long‐run unbiased expectations. These assumptions are unnecessary for interval‐coded data. In April 2004, the Monthly Consumer Confidence Survey in Japan started to ask households about their price expectations a year ahead in seven categories with partially known boundaries. Thus one can identify up to six parameters including an indifference interval each month. This paper compares normal, skew normal (SN), skew exponential power (SEP), and skew t (St) distributions, and finds that an St distribution fits the data well. The results help us to better understand the dynamics of heterogeneous expectations.

Skew scale mixtures of normal distributions: Properties and estimation

Scale mixtures of normal distributions are often used as a challenging class for statistical procedures for symmetrical data. In this article, we have defined a skewed version of these distributions and we have derived several of its probabilistic and inferential properties. The main virtue of the members of this family of distributions is that they are easy to simulate from and they also supply genuine EM algorithms for maximum likelihood estimation. For univariate skewed responses, the EM-type algorithm has been discussed with emphasis on the skew- t , skew-slash, skew-contaminated normal and skew-exponential power distributions. Some simplifying and unifying results are also noted with the Fisher information matrix, which is derived analytically for some members of this class. Results obtained from simulated and real data sets are reported, illustrating the usefulness of the proposed methodology. The main conclusion in reanalyzing a data set previously studied is that the models so far entertained are clearly not the most adequate ones.

On the singularity of multivariate skew-symmetric models

In recent years, the skew-normal models introduced by Azzalini (1985) [1]–and their multivariate generalizations from Azzalini and Dalla Valle (1996) [4]–have enjoyed an amazing success, although an important literature has reported that they exhibit, in the vicinity of symmetry, singular Fisher information matrices and stationary points in the profile log-likelihood function for skewness, with the usual unpleasant consequences for inference. It has been shown (DiCiccio and Monti (2004) [23], DiCiccio and Monti (2009) [24] and Gómez et al. (2007) [25]) that these singularities, in some specific parametric extensions of skew-normal models (such as the classes of skew- t or skew-exponential power distributions), appear at skew-normal distributions only. Yet, an important question remains open: in broader semiparametric models of skewed distributions (such as the general skew-symmetric and skew-elliptical ones), which symmetric kernels lead to such singularities? The present paper provides an answer to this question. In very general (possibly multivariate) skew-symmetric models, we characterize, for each possible value of the rank of Fisher information matrices, the class of symmetric kernels achieving the corresponding rank. Our results show that, for strictly multivariate skew-symmetric models, not only Gaussian kernels yield singular Fisher information matrices. In contrast, we prove that systematic stationary points in the profile log-likelihood functions are obtained for (multi)normal kernels only. Finally, we also discuss the implications of such singularities on inference.

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.