# Bayesian Reference Analysis for the Generalized Normal Linear Regression Model

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Normal Distribution

## 3. Generalized Normal Linear Regression Model

**Proposition**

**1.**

## 4. Objective Bayesian Analysis

**Proposition**

**2.**

**Corollary**

**1.**

#### 4.1. Reference Prior

#### 4.2. A Problem with the Jeffreys Prior

**Corollary**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 5. Metropolis–Hastings Algorithm

- Set the values ${\mathit{\omega}}^{\left(0\right)}=\left(\right)open="("\; close=")">{\omega}_{1}^{\left(0\right)},{\omega}_{2}^{\left(0\right)},{\omega}_{3}^{\left(0\right)}$.
- Generate ${\omega}_{1}^{*}$ from the proposal distribution $q({\omega}_{1}^{\left(0\right)},{\omega}_{1}^{*})$.
- Sample u from a uniform distribution $U(0,1)$.
- If $u\le min\left(\right)open="\{"\; close="\}">1,exp\left(\right)open="["\; close="]">log{\pi}^{\mathsf{R}}\left(\right)open="("\; close=")">{\omega}_{1}^{*}|{\omega}_{2}^{\left(0\right)},{\omega}_{3}^{\left(0\right)},\mathbf{y}$ then update ${\omega}_{1}^{\left(1\right)}$ from ${\omega}_{1}^{*}$; otherwise, use the value of ${\omega}_{1}^{\left(0\right)}$, i.e., ${\omega}_{1}^{\left(1\right)}={\omega}_{1}^{\left(0\right)}$.
- Repeat the same steps above for ${\omega}_{2}^{\left(1\right)}$ and ${\omega}_{3}^{\left(1\right)}$.
- Repeat Steps 2–5 until we obtain the target sample size.

## 6. Selection Criteria for Models

## 7. Bayesian Case Influence Diagnostics

## 8. Applications

#### 8.1. Artificial Data

#### 8.2. Real Data Set

## 9. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

**Proof of Corollary**

**2.**

**Proof of Corollary**

**3.**

**Proof of Proposition**

**3.**

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**Figure 1.**Density functions of the GN distribution with the parameters (

**a**) $\mu =0$ and $\sigma =1$ fixed and varying s and (

**b**) $\mu =0$ and $s=1.5$ fixed and varying $\sigma $.

**Figure 2.**Real data. Generalized normal linear regression model with $s=2.865$. (

**a**) Scatterplot of the data and fit generalized normal linear regression model. (

**b**) Quantile residuals versus fit values. (

**c**) Quantile residuals versus the index. (

**d**) Graph of the quantiles of the GN distribution for the residuals of the model.

**Figure 3.**Real data. Normal linear regression model. (

**a**) Scatterplot of the data and fit normal linear regression model. (

**b**) Quantile residuals versus fit values. (

**c**) Quantile residuals versus the index. (

**d**) Graph of the quantiles of the normal distribution for the residuals of the model.

**Table 1.**Artificial data. Posterior mean, median, standard deviation (SD), and $95\%$ HDP intervals for the parameters of the model.

Parameters | Mean | Median | SD | 95% HDP |
---|---|---|---|---|

${\mathit{\beta}}_{1}$ | $1.995$ | $1.996$ | $0.086$ | $(1.826;2.164)$ |

${\mathit{\beta}}_{2}$ | $-1.510$ | $-1.510$ | $0.032$ | $(-1.572;-1.447)$ |

$\sigma $ | $1.027$ | $1.028$ | $0.055$ | $(0.922;1.135)$ |

s | $2.657$ | $2.631$ | $0.320$ | $(2.042;3.293)$ |

**Table 2.**Artificial data. Posterior mean, $RV(\%)$, and $95\%$ HDP intervals for the parameters of the model.

Data Names | Perturbed Case | Parameters | Mean | RV (%) | 95% HDP |
---|---|---|---|---|---|

a | None | ${\beta}_{1}$ | $1.995$ | − | $(1.826;2.164)$ |

${\beta}_{2}$ | $-1.510$ | − | $(-1.572;-1.447)$ | ||

$\sigma $ | $1.027$ | − | $(0.922;1.135)$ | ||

s | $2.657$ | − | $(2.041;3.293)$ | ||

b | 50 | ${\beta}_{1}$ | $2.026$ | $1.554$ | $(1.841;2.211)$ |

${\beta}_{2}$ | $-1.517$ | $0.464$ | $(-1.585;-1.449)$ | ||

$\sigma $ | $0.970$ | $5.550$ | $(0.866;1.073)$ | ||

s | $2.188$ | $17.651$ | $(1.763;2.613)$ | ||

c | 250 | ${\beta}_{1}$ | $1.966$ | $1.454$ | $(1.775;2.154)$ |

${\beta}_{2}$ | $-1.495$ | $0.993$ | $(-1.564;-1.423)$ | ||

$\sigma $ | $0.916$ | $10.808$ | $(0.816;1.016)$ | ||

s | $1.867$ | $29.733$ | $(1.565;2.165)$ | ||

d | $\{50,250\}$ | ${\beta}_{1}$ | $2.002$ | $1.332$ | $(1.807;2.198)$ |

${\beta}_{2}$ | $-1.506$ | $0.265$ | $(-1.578;-1.436)$ | ||

$\sigma $ | $0.902$ | $12.171$ | $(0.801;1.003)$ | ||

s | $1.773$ | $33.271$ | $(1.496;2.052)$ |

Data Names | Case Number | $\mathit{K}(\mathit{\pi},{\mathit{\pi}}_{(-\mathit{i})})$ | Calibration |
---|---|---|---|

a | 50 | 0.0121 | 0.5774 |

250 | 0.0014 | 0.5262 | |

b | 50 | 16.1593 | 1.0000 |

c | 250 | 19.2236 | 1.0000 |

d | 50 | 2.8796 | 0.9992 |

250 | 18.2292 | 1.0000 |

**Table 4.**Real data. Posterior mean and $95\%$ HPD intervals for the parameters of the model and Bayesian comparison criteria.

Model | Parameters | Mean | 95% HDP | DIC | EBIC | B |
---|---|---|---|---|---|---|

${\beta}_{1}$ | 7.066 | (6.398; 7.724) | ||||

Generalized | ${\beta}_{2}$ | 0.948 | (0.907; 0.990) | 5829.180 | 5847.96 | −2914.595 |

Normal | $\sigma $ | 3.237 | (3.034; 3.440) | |||

s | 2.865 | (2.436; 3.305) | ||||

${\beta}_{1}$ | 7.002 | (6.331; 7.671) | ||||

Normal | ${\beta}_{2}$ | 0.949 | (0.907; 0.991) | 6248.652 | 6262.98 | −3126.614 |

$\sigma $ | 1.993 | (1.916; 2.068) |

Case Number | $\mathit{K}(\mathit{\pi},{\mathit{\pi}}_{(-\mathit{i})})$ | Calibration |
---|---|---|

335 | 0.1683 | 0.7673 |

479 | 0.1734 | 0.7707 |

803 | 0.4936 | 0.8960 |

**Table 6.**Posterior estimates and $RV(\%)$ for Eucalyptus height and diameter data following the removal of the influential case.

Cases Deletions | Parameter | Mean | $\mathit{RV}(\%)$ | 95% HDP |
---|---|---|---|---|

803 | ${\beta}_{1}$ | 7.067 | 0.014 | (6.375; 7.753) |

${\beta}_{2}$ | 0.948 | - | (0.905; 0.991) | |

$\sigma $ | 3.252 | 0.463 | (3.054; 3.454) | |

s | 2.936 | 2.478 | (2.486; 3.390) |

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**MDPI and ACS Style**

Tomazella, V.L.D.; Jesus, S.R.; Gazon, A.B.; Louzada, F.; Nadarajah, S.; Nascimento, D.C.; Rodrigues, F.A.; Ramos, P.L.
Bayesian Reference Analysis for the Generalized Normal Linear Regression Model. *Symmetry* **2021**, *13*, 856.
https://doi.org/10.3390/sym13050856

**AMA Style**

Tomazella VLD, Jesus SR, Gazon AB, Louzada F, Nadarajah S, Nascimento DC, Rodrigues FA, Ramos PL.
Bayesian Reference Analysis for the Generalized Normal Linear Regression Model. *Symmetry*. 2021; 13(5):856.
https://doi.org/10.3390/sym13050856

**Chicago/Turabian Style**

Tomazella, Vera Lucia Damasceno, Sandra Rêgo Jesus, Amanda Buosi Gazon, Francisco Louzada, Saralees Nadarajah, Diego Carvalho Nascimento, Francisco Aparecido Rodrigues, and Pedro Luiz Ramos.
2021. "Bayesian Reference Analysis for the Generalized Normal Linear Regression Model" *Symmetry* 13, no. 5: 856.
https://doi.org/10.3390/sym13050856