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

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## 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.

## Keywords

Bayesian inference Loss-based prior Objective Bayes Electricity prices## 1 Introduction

Two-piece location-scale models have been mainly used for modeling data exhibiting departures from symmetry. Moreover, some specific two-piece location-scale distributions have been employed in finance to represent the errors in GARCH-type models, see Zhu and Zinde-Walsh (2009), Zhu and Galbraith (2011). Different mechanisms have been presented to obtain skewed distributions by modifying symmetric distributions (Azzalini 1985; Fernandez and Steel 1998; Mudholkar and Hutson 2000). Recently, the objective Bayesian literature focused on this class of models. Firstly, Rubio and Steel (2014) derived the Jeffreys rule prior and the independence Jeffreys priors for different families of skewed distributions. They show that Jeffreys priors for some distributions, such as the skewed Student-*t*, lead to improper posterior distributions. Conversely, reference priors have shown to be more suitable for the above class of distributions, see Tu et al. (2016).

In this work, we introduce a novel objective prior for some distributions of the class of two-piece location-scale models, such as the skewed exponential power distribution (SEPD) and the skewed generalized logistic distribution (SGLD). Recently, SEPD has been introduced in regression and stochastic volatility models by Naranjo et al. (2015) and Kobayashi (2016). Moreover, SEPD has been used as working likelihood in the quantile regression analysis by Bernardi et al. (2018) and for Bayesian Conditional Autoregressive Risk Measures by Bernardi et al. (2019). Following Leisen et al. (2017), we introduce a Bayesian approach obtained by applying the loss-based prior discussed in Villa and Walker (2015). In particular, we derive the loss-based prior for the parameter that controls heaviness of the tails of the distribution.

In the literature, the asymmetric Laplace distribution (ALD) or the asymmetric Student-*t* 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 Sect. 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 Sect. 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 Sect. 6.

## 2 Two-piece location-scale models

*f*is an absolutely continuous distribution on \(\mathbb {R}\), \(\mu \in \mathbb {R}\) is the location parameter, \(\sigma _1 \in \mathbb {R}^+\) and \(\sigma _2 \in \mathbb {R}^+\) are the separate scale parameters and \(\alpha \in (0,1)\) is the skewness parameter. In this paper, we follow Rubio and Steel (2014) and assume

*f*to be symmetric with a single mode at zero, which means that \(\mu \) is the mode of the density in (1). Hereafter, we assume \(\sigma _1=\sigma _2=\sigma \) and we focus on three particular two-piece location-scale models: the skewed Student-

*t*distribution (SST), the skewed exponential power distribution (SEPD) and the skewed generalized logistic distribution (SGLD). These distributions depend on an additional parameter

*p*which controls the behaviour of the tails. The SST is defined as follows.

### Definition 2.1

*Skewed Student-*

*t*

*distribution*) Assume \(\mu \in \mathbb {R}\) the location parameter, \(\sigma >0\) the scale parameter, \(\alpha \in (0,1)\) the skewness parameter and \(p>0\) the tail parameter. We define the skewed Student-t distribution as

*p*.

For a more detailed description of the properties of the SST distribution, see Fernandez and Steel (1998) and Zhu and Galbraith (2010). The SST has some special cases: if \(\alpha =1/2\), it is the usual Student-*t* with *p* degrees of freedom; if \(p=1\), is the skewed Cauchy, while for \(p\rightarrow \infty \), it converges to the skewed normal distribution. Leisen et al. (2017) have proposed a loss-based prior for the tail parameter *p* of the SST distribution. Therefore, hereafter we will give limited attention to this distribution and we will focus on the remaining two distributions.

The first distribution of interest in our analysis accommodates heavy tails as well as skewness and is defined as follows.

### Definition 2.2

*Skewed exponential power distribution*) Let us define \(\mu \in \mathbb {R}\), the location parameter, \(\sigma >0\), the scale parameter, \(\alpha \in (0,1)\) the skewness parameter and \(p>0\) the tail parameter. The skewed exponential power distribution has the form

The SEPD has been studied in Fernandez and Steel (1998), Komunjer (2007) and Zhu and Zinde-Walsh (2009). In detail, for \(p=1\) the SEPD becomes a skewed Laplace distribution, and for \(p=2\) is a skewed normal distribution. For values of \(p\rightarrow \infty \), we have that the SEPD reduces to an uniform distribution.

*p*,

*p*) and

*S*(

*x*) and

*s*(

*x*) are, respectively, the cumulative distribution function and the probability density function of the logistic distribution

*f*(

*x*) in (4) is also known as the logistic distribution of the III type. Its skewed version is defined as follows.

### Definition 2.3

*Skewed generalized logistic distribution*) Assume \(\mu \in \mathbb {R}\), the location parameter, \(\sigma >0\), the scale parameter, \(\alpha \in (0,1)\) the skewness parameter and \(p>0\) the tail parameter. We define the skewed generalized logistic distribution as

*B*(

*p*,

*p*) is the beta function.

## 3 The objective prior distribution

*p*does not depend on \(\alpha \), \(\mu \) and \(\sigma \). As such, we can write \(\pi (p|\alpha ,\mu ,\sigma )=\pi (p)\).

### 3.1 Loss-based prior for *p*

The main focus of this paper is to make inference on the parameter *p*. Without loss of generality, *p* is considered discrete taking values in \(\mathbb {N}\). This is motivated by the fact that seldom the amount of information about *p* in the data is sufficient to discern between distributions that differ in *p* less than one. For instance, this is a well known fact for the Student-*t* distribution.

*p*. The idea is to assign a

*worth*to each parameter value by objectively measuring what is lost if the value is removed, and it is the true one. The loss is evaluated by applying the well known result in Berk (1966) stating that, if a model is misspecified, the posterior distribution asymptotically accumulates on the model which is the nearest to the true one, in terms of the Kullback–Leibler divergence. Therefore, the

*worth*of the parameter value

*p*is represented by the Kullback–Leibler divergence \(D_{\text {KL}}\left( f_p^{\alpha ,\mu ,\sigma }\Vert f_{p'}^{\alpha ,\mu ,\sigma }\right) \), where \(p^\prime \ne p\) is the parameter value that minimizes the divergence. To link the

*worth*of a parameter value to the prior mass, Villa and Walker (2015) use the self-information loss function. This particular type of loss function measures the loss in information contained in a probability statement (Merhav and Feder 1998). As we now have, for each value of

*p*, the loss in information measured in two different ways, we simply equate them obtaining the loss-based prior:

Following Leisen et al. (2017), we introduce a theorem (which proof is in “Appendix A”) to study the form of the Kullback–Leibler divergence and consequently of the loss-based prior for the tail parameter *p*.

### Theorem 3.1

In other words, Theorem 3.1 shows that the loss-based prior distribution for the tail parameter *p* does not depend from the skewness parameter \(\alpha \), the location \(\mu \) and the scale \(\sigma \). Hence, the prior can be written as \(\pi (p|\alpha ,\mu ,\sigma ) = \pi (p)\).

The following theorem derives the closed form of the Kullback–Leibler divergence for the SEPD. Its proofs, which can be found in “Appendix A”, leveraged on the result in Theorem 3.1.

### Theorem 3.2

*p*is:

*p*is proper for \(p=\{1,2,3,\ldots ,\infty \}\), therefore yielding a proper posterior.

To derive the loss-based prior for the parameter *p* of the SGLD, we consider the following Theorem 3.3 (which proof is in the “Appendix A”), giving the expression of the Kullback–Leibler divergence between two SGLDs.

### Theorem 3.3

### 3.2 Non-informative prior for the parameters \(\alpha \), \(\mu \) and \(\sigma \).

In line with the minimally informative focus of the paper, we have selected objective priors for the other parameters of the considered distributions. That is, we have considered Jeffreys priors for \(\alpha \), \(\mu \) and \(\sigma \). As mentioned at the beginning of Sect. 3, we assume that the prior information on the true value of the parameters is independent. As such, we can consider, not only \(\pi (\alpha )\) on its own, but we can also factorise the prior of the location and the scale parameters; that is \(\pi (\mu ,\sigma )=\pi (\mu )\pi (\sigma )\). The Jeffreys prior for \(\mu \) and \(\sigma \) is then proportional to \(1/\sigma \), which is obtained by considering the Jeffreys prior for a location parameter, \(\pi (\mu )\propto 1\), and the Jeffreys prior for a scale parameter, \(\pi (\sigma )\propto 1/\sigma \). Both these priors are extensively discussed in Jeffreys (1961). It is worthwhile to note that the above considerations recover the well-known reference prior for the pair \((\mu ,\sigma )\) (Berger et al. 2009).

Finally, the Jeffreys prior for the skewness parameter \(\alpha \) has been introduced in Rubio and Steel (2014), and it shows to be a Beta distribution with both parameters equal to 1 / 2. That is \(\pi (\alpha )\sim \text{ Be }(1/2,1/2)\).

## 4 Simulation studies

It is important to analyse the performances of objective priors by studying the frequentist properties of the posterior distributions they yield to. As such, the aim of this section is to present simulation studies concerning the objective priors for *p*, as defined in Sect. 3, for the considered two-piece location-scale models discussed in this work. In particular, we study time series where the residual error terms follow a SEPD and regression models with error terms that follow a SGLD.

### 4.1 SEPD simulation study

*p*as defined in Sect. 3.1. The prior for the remaining three parameters of the SEPD, has been fixed as explained in Sect. 3.2. For the parameter \(\phi _1\), we assume a Zellner prior (Zellner 1986) with \(g=T\), that is \(N(0,T(\sum _{i=1}^{T-1} y_i^2)^{-1})\). For each of the above scenarios, we have generated 250 random samples, as described in the “Appendix C”, and computed the frequentist coverage of the 95% posterior credible interval for

*p*, and the relative square root of the mean squared error \(\sqrt{\text{ MSE }(p)}/p\). The coverage measures the frequency of which the true parameter value for

*p*is included in the 95% credible interval of the posterior distribution of the parameter. Ideally, this value should be close to 0.95. The MSE allows to have a measure of the accuracy of the estimate, intended as the posterior mean for

*p*.

As the yielded posterior distribution for the parameters is not analytically tractable, it is necessary to adopt Markov Chain Monte Carlo (MCMC) methods. In particular, we have implemented a Metropolis within Gibbs sampler. For each of the above 250 samples, we have run 20, 000 iterations of the MCMC algorithm and discarded the first 5000 iterations as burn-in period. The results of the frequentist analysis of the posterior of *p* are plotted in Fig. 1. Examining the coverage, we note that the samples with \(T=100\) have a frequency closer to the nominal value (i.e. 95%) compared to the samples with \(T=250\); this is more obvious for relatively large values of *p*. The MSE behaves in line with other frequentist studies for tail parameters (such as for the Student-*t* and the skewed Student-*t*), with a smaller index value for larger sample size (as expected). Finally, we note that the effect of \(\alpha \) on the frequentist performances is negligible.

*p*and \(\sigma \). The corresponding posterior mean, median and 95% HPD credible set are reported in Table 1. We note that the true parameter values are well contained in the corresponding posterior credible interval.

### 4.2 SGLD simulation study

*p*of the SGLD, as anticipated, we consider a linear regression model where the error terms have the above distribution. That is,

*p*, the Jeffreys prior for the skewness parameter, \(\pi (\alpha )\sim \text{ Be }(1/2,1/2)\), and for the scale parameter, \(\pi (\sigma )\propto 1/\sigma \) (as discussed in Sect. 3.2). For \(\beta _0\) and \(\beta _1\) we have used the Zellner g-prior (Zellner 1986) with \(g=n\), which is a bivariate normal with zero means and covariance matrix \(\Sigma =n(X^\prime X)^{-1}\), where \(X = (1, x_{1i})\).

*p*is shown in Fig. 3. The coverage of the posterior 95% credible interval appears to be very similar whether we consider the different values of the skewness parameter \(\alpha \) or the sample size. For what it concerns the MSE, we note some differences when the sample size is 30, although these are most certainly due to the relatively small amount of information about

*p*contained in the sample. This difference vanishes for \(n=100\). Similarly to the study of the SEPD model, we report the complete inferential procedure for a single sample drawn from the model in (13), where we have set \(\beta _0=-2.5\), \(\beta _1=3\), \(\alpha =0.23\), \(p=9\) and \(\sigma =1\). We run an MCMC procedure with 30.000 iterations and a burn-in period of 5.000 iterations. The posterior chains and histograms are plotted in Fig. 4, with the corresponding posterior statistics reported in Table 2. We note that the posterior means and medians give an excellent point representation of the true parameter values, and that the posterior credible intervals contain the above true values giving a high level accuracy of the estimates.

Summary statistics of the posterior distributions for the parameters of the simulated data from an SEPD with \(\alpha = 0.23\), \(\phi _1 = -0.5\), \(p=9\), \(\sigma =1\) and \(T=300\)

Parameter | Mean | Median | \(95\%\) HPD |
---|---|---|---|

\(\alpha \) | 0.2325 | 0.2310 | (0.2144, 0.2542) |

\(\phi _1\) | \(-\) 0.4672 | \(-\) 0.4684 | (\(-\) 0.5180, \(-\) 0.4091) |

| 9.5238 | 9 | (5, 17) |

\(\sigma \) | 1.0231 | 1.0214 | (0.9033,1.1533) |

## 5 Real data analysis

In this section, we present two different examples with publicly available data to illustrate how the loss-based prior for the tail parameter *p* performs. In the first example we analyse the Nordpool Electricity prices by means of an autoregressive model with error terms distributed as a skewed exponential power, while in the second example we apply a linear regression model with error terms distributed as a skewed generalised logistic to Small Cell Cancer data.

### 5.1 Nordpool electricity prices data

The data is modelled with a univariate autoregressive model with one lag, where the error terms are SEPD. The results of the analysis are based on one-step-ahead forecasting process with a rolling window approach of 10 years for both the countries, and we have a forecast evaluation period of 60 observations (from January 2013 to December 2017). Following the results in Sect. 4, we run the estimation procedure through Gibbs sampling with a burn-in of 5.000 iterations and for the forecasting procedure we use the remaining 15.000 iterations.

*T*is the number of observations,

*R*is the length of the rolling window and \({\hat{y}}_{t+1|t}\) are the price forecasts.

Summary statistics of the posterior distributions for the parameters of the simulated data from an SGLD with \(\alpha = 0.13\), \(\beta _0 = -2.5\), \(\beta _1 = 3\), \(p=9\) and \(n=300\)

Parameter | Mean | Median | \(95\%\) HPD |
---|---|---|---|

\(\alpha \) | 0.1363 | 0.136 | (0.1155,0.1615) |

\(\beta _0\) | \(-\) 2.5396 | \(-\) 2.54 | (\(-\) 2.5775, \(-\) 2.4964) |

\(\beta _1\) | 3.0048 | 3.0041 | (2.9639, 3.0494) |

| 8.9295 | 9 | (8, 10) |

*t*. In addition, following Gneiting and Raftery (2007) and Gneiting and Ranjan (2011), we also compute the continuous ranked probability score, which has some advantages with respect to the log-score. In fact, it is less sensitive to outliers. It can be computed as follows:

*F*denotes the cumulative distribution function associated with the predictive density

*f*, \(\mathbb {I}\{y_{t+1}\le z\}\) denotes an indicator function taking value 1 if \(y_{t+1}\le z\) and 0 otherwise, and \(Y_{t+1}\) and \(Y'_{t+1}\) are independent random draws from the posterior predictive density.

Point (RMSE) and density forecast (average log predictive score and average CRPS) for Finland and Denmark

Forecast | OLS | Bayesian Normal | SEPD error | |
---|---|---|---|---|

| RMSE | 0.749 | 1.003 | 1.022 |

log-score | \(-\) 1.334 | 0.090 | 0.106 | |

CRPS | 0.491 | 0.899 | 0.891 | |

| RMSE | 0.541 | 1.001 | 1.021 |

log-score | \(-\) 1.143 | 0.013 | 0.165 | |

CRPS | 0.355 | 0.990 | 0.918 |

For both Finland and Denmark the point forecast appears to be worse than the benchmark. This is more obvious for the AR model with SEPD errors, although the values are not far from one. There is a noticeable improvement, in using SEPD errors, when we focus on density forecast. In fact, considering the log-score, we have a improvement in considering SEPD (instead of normal) errors from 0.090 to 0.106 for Finland, and a more obvious improvement from 0.013 to 0.165 for Denmark.

### 5.2 Small cell cancer data

In this second example we illustrate the loss-based prior for the tail parameter *p* when we employ a linear regression model with SGLD errors. The data has been obtained from Ying et al. (1995), where a lung cancer study with two different types of treatment has been performed. In particular, the study contained \(n=121\) survival times (in log-days) of patients with small cell lung cancer (SCLC) to whom were administrated two different therapies. A treatment consisted of a combination of etoposide (E) and cisplatin (P) in any order. The patient were split into two treatment groups: treatment A (62 patients), where the therapy consisted in administering P followed by E; treatment B (59 patients), where the therapy consisted in administering E followed by P. We regress the survival time on the following two covariates: the entry age (in years) and a dummy variable identifying the type of treatment (A or B).

The estimation of the parameters of the regression model has been done through Monte Carlo methods, as described in Sect. 4, with 50,000 iterations and a burn-in period of 10,000 iterations. Figure 6 shows the histograms of the posterior distributions for the parameters, while in Table 4 we have the corresponding posterior statistics.

SCLC Lung Cancer data: Posterior mean, posterior median and \(95\%\) HPD credible set of the posterior for the regression model parameters

Parameter | Mean | Median | \(95\%\) HPD | Rubio & Yu |
---|---|---|---|---|

Intercept | 6.8438 | 6.9059 | (5.6692, 7.4672) | 6.690 |

Entry age | \(-\) 0.0105 | \(-\) 0.0117 | (\(-\) 0.0216, 0.0079) | \(-\) 0.009 |

Treatment | \(-\) 0.3637 | \(-\) 0.3611 | (\(-\) 0.7161, \(-\) 0.0552) | \(-\) 0.446 |

\(\alpha \) | 0.3837 | 0.3814 | (0.2717, 0.5116) | \(-\) 0.395 (\(\gamma \)) |

| 4.5349 | 3 | (1, 14) | NA |

\(\sigma \) | 0.8630 | 0.7771 | (0.3649, 1.7209) | 0.650 |

## 6 Discussion

We have illustrated an objective Bayesian approach in the estimation of the tail parameter in two particular distributions: the skewed exponential power distribution (SEPD) and the skewed generalised logistic distribution (SGLD). This represents a new application of the well-known loss-based prior (Villa and Walker 2015), where information theoretical considerations are used to derive minimally informative prior distributions. The SEPD and the SGLD are part of the wider family of two-piece location-scale distribution and allow to entangle skewness and tail fatness in one single probability distribution. Therefore, they represent an appealing modeling solution in scenarios where such behaviours are exhibited by the data, such as in financial applications and survival analysis.

We illustrate the properties of the loss-based prior for the tail parameter of the above distributions by performing a thorough simulation study and analysis two real data sets. Furthermore, we show how the SEPD and SGLD can be used to model error terms in complex modeling situations, such as the error terms of autoregressive process for time series and error terms for linear regression models.

We conclude the paper with the indication of some future research lines. Our approach can be extended to volatility models in order to model the tail behaviour of the returns in electricity markets. Furthermore, a combination of our loss-based approach with quantile regression could be a possible extension.

## Notes

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