Bayesian analysis of some models that use the asymmetric exponential power distribution

Abstract

The asymmetric exponential power (AEP) family includes the symmetric exponential power distribution as a particular case. It provides flexible distributions with lighter and heavier tails compared to the normal one. The distributions of this family can successfully handle both symmetry/asymmetry and light/heavy tails simultaneously. Even more, the distributions can fit each tail separately. This provides a great flexibility when fitting experimental data. The idea of using a scale mixture of uniform representation of the AEP distribution is exploited to derive efficient Gibbs sampling algorithms in three different Bayesian contexts. Firstly, a posterior exploration is performed, where the AEP distribution is considered for the likelihood model. Secondly, a linear regression model, that uses the AEP distribution for the error variable, is developed. And finally, a binary regression model is analyzed, by using the inverse of the AEP cumulative distribution function as the link function. These three models have been built in such a way that they share some full conditional distributions to sample from their respective posterior distributions. The theoretical results are illustrated by comparing with other competing models using some previously published datasets.

Appendices

Appendix 1. Proofs

Proposition 1.

Proof

It is enough to show that

$$\begin{aligned}&\int f(y|u_1)f(u_1)\mathrm {d}u_1 \\&\quad = \int \tfrac{1}{\alpha \sigma }\exp (-u_1) I\Big [\mu -\tfrac{\alpha \sigma }{\varGamma (1+1/\theta _1)}u_1^{1/\theta _1}<y\le \mu \Big ]\\&\quad \qquad I[u_1>0] \mathrm {d}u_1\\&\quad = \int \tfrac{1}{\alpha \sigma }\exp (-u_1) I\Big [u_1>\big |\tfrac{y-\mu }{\alpha \sigma /\varGamma (1+1/\theta _1)}\big |^{\theta _1}\Big ] I[y\le \mu ] \mathrm {d}u_1\\&\quad =\tfrac{1}{\alpha \sigma } \exp \Big (-\big |\tfrac{y-\mu }{\alpha \sigma /\varGamma (1+1/\theta _1)}\big |^{\theta _1}\Big ) I[y\le \mu ],\\&\int f(y|u_2)f(u_2)\mathrm {d}u_2 \\&\quad = \int \tfrac{1}{(1-\alpha )\sigma }\exp (-u_2) I\Big [\mu <y<\mu +\tfrac{(1-\alpha )\sigma }{\varGamma (1+1/\theta _2)}u_2^{1/\theta _2}\Big ]\\&\quad \qquad I[u_2>0] \mathrm {d}u_2\\&\quad = \int \tfrac{1}{(1-\alpha )\sigma }\exp (-u_2) I\Big [u_2>\big |\tfrac{y-\mu }{(1-\alpha )\sigma /\varGamma (1+1/\theta _2)}\big |^{\theta _2}\Big ]\\&\quad \qquad I[y>\mu ] \mathrm {d}u_2\\&\quad = \tfrac{1}{(1-\alpha )\sigma } \exp \Big (-\big |\tfrac{y-\mu }{(1-\alpha )\sigma /\varGamma (1+1/\theta _2)}\big |^{\theta _2}\Big ) I[y>\mu ], \end{aligned}$$

and so

$$\begin{aligned} f(y)&= \alpha \int f(y|u_1)f(u_1)\mathrm {d}u_1\\&+ (1-\alpha )\int f(y|u_2)f(u_2)\mathrm {d}u_2 \end{aligned}$$

is the pdf of the distribution $\mathrm {AEP}(\mu ,\sigma ,\alpha ,\theta _1,\theta _2)$.

Appendix 2. Specific full conditional distributions

Equation (3):

If $\pi (\mu )\propto 1$ this distribution becomes an uniform

$$\begin{aligned}{}[\mu | \mathbf {y},\mathbf {u}_1,\mathbf {u}_2 , \sigma ,\alpha , \theta _1,\theta _2] \sim \mathrm {U}\big ( \underline{\mu }\,,\,\overline{\mu } \big ). \end{aligned}$$

If $\pi (\mu )$ is the pdf of a normal distribution, $\mathrm {N}(m_{\mu },s^2_{\mu }),$ the full conditional distribution becomes a truncated normal

$$\begin{aligned}{}[\mu | \mathbf {y},\mathbf {u}_1,\mathbf {u}_2 , \sigma ,\alpha , \theta _1,\theta _2] \sim \mathrm {N}(m_{\mu },s^2_{\mu })I\big [ \underline{\mu } < \mu < \overline{\mu } \big ]. \end{aligned}$$

Equation (4):

If $\pi (\sigma )\propto 1$, the full conditional distribution of $\sigma$ is

$$\begin{aligned}&[\sigma | \mathbf {y}, \mathbf {u}_1,\mathbf {u}_2, \mu , \alpha , \theta _1,\theta _2]\\&\quad \sim \mathrm {ParetoI}\big ( \text {scale}=\underline{\sigma }\,,\,\text {shape}=n-1\big ). \end{aligned}$$

If $\pi (\sigma )\propto 1/\sigma ^{m_{\sigma }}$, the full conditional distribution of $\sigma$ is

$$\begin{aligned}&[\sigma |\mathbf {y}, \mathbf {u}_1,\mathbf {u}_2, \mu , \alpha , \theta _1,\theta _2]\\&\quad \sim \mathrm {ParetoI}\big ( \text {scale}=\underline{\sigma }\,,\,\text {shape}=n-1+m_{\sigma }\big ). \end{aligned}$$

If $\pi (\sigma )$ is the pdf of an inverse-gamma distribution, $\mathrm {InvGamma}(a_{\sigma },b_{\sigma })$, the full conditional distribution of $\sigma$ is

$$\begin{aligned}&[\sigma | \mathbf {y}, \mathbf {u}_1,\mathbf {u}_2, \mu , \alpha , \theta _1,\theta _2]\\&\quad \sim \mathrm {InvGamma} \big (\text {shape}\!=\!n-1\!+\!a_{\sigma }, \text { scale}=b_{\sigma }\big ) I\big [\sigma \!>\! \underline{\sigma }\big ]. \end{aligned}$$

Equation (5):

If $\pi (\alpha )\propto 1$, the full conditional distribution of $\alpha$ is

$$\begin{aligned}{}[\alpha | \mathbf {y}, \mathbf {u}_1,\mathbf {u}_2, \mu , \sigma , \theta _1,\theta _2] \sim \mathrm {U}\big (\underline{\alpha }\,,\, \overline{\alpha } \big ). \end{aligned}$$

If $\pi (\alpha )$ is the pdf of a beta distribution, $\mathrm {Beta}(a_{\alpha },b_{\alpha })$, the full conditional distribution of $\alpha$ is

$$\begin{aligned}{}[\alpha | \mathbf {y}, \mathbf {u}_1,\mathbf {u}_2, \mu , \sigma , \theta _1,\theta _2] \sim \mathrm {Beta}(a_{\alpha }\ ,\ b_{\alpha }) I\big [\underline{\alpha } < \alpha < \overline{\alpha } \big ]. \end{aligned}$$