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

Abstract

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.MSC: 62F15, 62M10.In the literature, the asymmetric Laplace distribution (ALD) or the asymmetric Studentt distribution (AST) have gained importance in a wide range of disciplines, such as economics (Zhao et al., 2007;Leisen et al., 2017), financial analysis (Zhu and Galbraith, 2010;Kozubowski and Podgorski, 2001;Harvey and Lange, 2016) and microbiology (Rubio and Steel, 2011). However, the application of the SEPD and SGLD to represent the errors of time series and regression models, has received limited attention in the context of objective Bayesian analysis. The aim of this paper is to contribute to the above research area by introducing an information theoretical approach to address inference on the tail parameter of the two skewed distributions.As currently there is a growing interest in electricity prices (see Weron (2014) and Nowotarski and Weron (2018) for a review), we will contribute to the analysis of monthly electricity prices in the Nordpool market, in particular for Denmark and Finland through an autoregressive model with errors distributed as a SEPD. Compared to the standard frequentist autoregressive approach, which is the benchmark in the literature (see Conejo et al. (2005), Misiorek et al. (2006 and Maciejowska and Weron (2015)), we can show that our methodology improves the density forecasting. In addition, we consider a linear regression model where the residuals are SGLD with a loss-based prior on the tail parameter. We illustrate the above model by studying the Small Cell Cancer data set in Ying et al. (1995) and in Rubio and Yu (2017).The structure of this document is as follows. In Section 2 we introduce the general two-piece location-scale distribution and discuss special distributions further developed in the paper, such as the SEPD and the SGLD. Section 3 focuses on the derivation of the objective priors for the parameters of the models here considered. In Section 4 we analyse the frequentist properties of the proposed prior using data simulated from regression models and time series models. Section 5 deals with real data, in particular we model electricity prices and a cancer dataset. Final discussion points and conclusions are presented in Section 6.

Bayesian estimation of electricity price risk with a multi-factor mixture of densities

“…In econometrics, one of the earliest contributions was by Hinkley and Revankar (1977). Jones (2015) and Villa et al (2019), as well as many references therein, discuss several families of skewed distributions and their applications.…”

Spatio-temporal variation in sea state parameters along virtual ship route paths

“…Graphical illustrations of the statistics are given by the PDFs shown in Figures 8 and 9 applicable to ∆ = 1h and ∆ = 3h, respectively. It can be seen that the empirical PDFs are fitted with two parametric PDFs; the normal and the t location-scale distributions, where the latter is also sometimes known as the Skewed Student t-distribution [39,40]. By nature, the normal distribution could be expected to be a good fit, but it is seen that it is not; in particular in case of the difference in mean wave direction.…”