# Parametric Quantile Regression Models for Fitting Double Bounded Response with Application to COVID-19 Mortality Rate Data

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## Abstract

**:**

## 1. Introduction

## 2. The Generalized Johnson ${\mathit{S}}_{\mathit{B}}$ Distribution

- We note that $\psi ={Q}^{-1}\left(\frac{{x}_{q}^{*}\left(\alpha \right)-\gamma}{\delta}\right)$ is the $100\times q$th quantile for the PGJSB model, where ${x}_{q}^{*}\left(\alpha \right)={G}^{-1}\left({q}^{1/\alpha}\right)$. Based on this idea, we also can reparametrize the model defining $\gamma ={x}_{q}^{*}\left(\alpha \right)-\delta Q\left(\psi \right)$. The pdf for this reparametrization is$$f(y;\psi ,\delta ,\alpha )=\delta \alpha {\left[G(\delta [Q\left(y\right)-Q\left(\psi \right)]+{x}_{q}^{*}\left(\alpha \right))\right]}^{\alpha -1}g(\delta [Q\left(y\right)-Q\left(\psi \right)+{x}_{q}^{*}\left(\alpha \right)]\left|\frac{\mathrm{d}Q\left(y\right)}{\mathrm{d}y}\right|,\phantom{\rule{1.em}{0ex}}y\in (0,1).$$In this work, we will refer to this specific parametrization as RPGJSB1${}_{q}(\psi ,\delta ,\alpha )$.
- Although $\alpha $ is a parameter that needs to be estimated from the sample, we can consider $\alpha \left(q\right)=-log\left(q\right)/log\left(2\right)$, $q\in (0,1)$ as fixed. With this definition, the cdf in (1) evaluated in $\xi $ is given by $F(\xi ;\xi ,\delta )={(1/2)}^{\alpha \left(q\right)}=q$. Therefore, fixing $\alpha \left(q\right)=-log\left(q\right)/log\left(2\right)$, $q\in (0,1)$, we have that $\xi $ represents the $100\times q$th quantile of the distribution; further, as in the work of Lemonte and Bazán [9], $\delta $ also can be interpreted as a dispersion parameter. We will refer to this parametrization as RPGJSB2${}_{q}(\xi ,\delta )$.

## 3. The Inference and Its Associated Diagnostic Analysis

#### 3.1. Inference

#### 3.2. Residuals

#### 3.3. Local Influence

#### 3.3.1. Perturbation of the Case Weights

#### 3.3.2. Perturbation of the Response

#### 3.3.3. Perturbation of the Predictor

## 4. Simulation Study

## 5. Data Analysis

#### 5.1. COVID-19 Data Set

`mort`: mortality rate (reported deaths/reported cases since the pandemic started). Mean = 0.020, Median = 0.018, standard deviation = 0.013, minimum = 0.003, and maximum = 0.092.`surface`: area of the country (in km${}^{2}$).`population`: official estimated population of the country.`cont`: continent to which the country belongs (categorized as 1: Africa, Asia, or Oceania; 2: the Americas; 3: Europe; with 69, 29, and 39 countries, respectively. This categorization was based on our previous analysis).

`surface`) and log(

`population`) separated by

`cont`.

#### 5.1.1. Estimation

`surface`) and

`America`were significant for modeling the quantile (with a nominal level of 5%) for all the considered q. This can be explained because countries with larger areas may have greater difficulties in providing medical coverage to their inhabitants compared to countries with smaller areas, and some countries in the Americas have been hit hard by the pandemic. The coefficients related to log(

`population`) and

`Europe`were significant for lower quantiles and not significant for higher quantiles.

`America`and

`Europe`decreased when q was increased. Furthermore, the coefficients related to the quantile for $log\left(\mathtt{population}\right)$ and $log\left(\mathtt{surface}\right)$ and the coefficients related to the scale of

`America`and

`Europe`remained similar for all q. Figure 4 presents the estimated $0.05,0.25,0.50,0.75$, and $0.95$ quantiles for the mortality rate for different values of $log\left(\mathtt{surface}\right)$.

#### 5.1.2. Local Influence Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AD | Anderson–Darling (test) |

AIC | Akaike information criterion |

BIC | Bayesian information criterion |

cdf | cumulative distribution function |

CP | coverage probabilities |

CVM | Cramér–Von-Mises (test) |

GJS | generalized Johnson ${S}_{B}$ distribution |

KS | Kolmogorov–Smirnov (test) |

LD | likelihood displacement |

ML | maximum likelihood |

probability distribution function | |

PJSB | power Johnson ${S}_{B}$ distribution |

PGJSB | power generalized Johnson ${S}_{B}$ distribution |

PGJSB1${}_{q}$ | model 1 with reparametrization power generalized Johnson ${S}_{B}$ distribution |

PGJSB2${}_{q}$ | model 2 with reparametrization power generalized Johnson ${S}_{B}$ distribution |

RQRs | randomized quantile residuals |

SW | Shapiro–Wilks (test) |

## Appendix A. Details of Local Influence

#### Appendix A.1. Perturbation of the Response

#### Appendix A.2. Perturbation of the Predictor

## Appendix B. COVID-19 Data Set

#### Appendix B.1. AIC and BIC Criteria

**Table A1.**AIC and BIC criteria for the RPGJSB1${}_{q}$ model of the COVID-19 data set considering 3 options for G (normal, logistic, and Cauchy) and the 4 discussed link functions.

Normal | Logistic | Cauchy | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Criteria | $\mathit{q}$ | Logit | Probit | Loglog | Cloglog | Logit | Probit | Loglog | Cloglog | Logit | Probit | Loglog | Cloglog |

0.05 | −871.9 | −871.7 | −871.5 | −871.9 | −871.4 | −871.3 | −871.3 | −871.3 | −840.3 | −841.5 | −842.4 | −840.1 | |

0.10 | −871.8 | −871.6 | −871.5 | −871.8 | −871.2 | −871.2 | −871.2 | −871.2 | −839.9 | −841.0 | −842.0 | −839.7 | |

0.15 | −871.7 | −871.6 | −871.5 | −871.8 | −871.2 | −871.2 | −871.1 | −871.2 | −839.7 | −840.8 | −841.8 | −839.5 | |

0.20 | −871.7 | −871.6 | −871.4 | −871.7 | −871.1 | −871.1 | −871.1 | −871.1 | −839.6 | −840.7 | −841.6 | −839.4 | |

0.25 | −871.6 | −871.5 | −871.4 | −871.7 | −871.1 | −871.1 | −871.1 | −871.1 | −839.5 | −840.6 | −841.6 | −839.3 | |

0.30 | −871.6 | −871.5 | −871.4 | −871.6 | −871.1 | −871.1 | −871.1 | −871.0 | −839.4 | −840.6 | −841.5 | −839.2 | |

0.35 | −871.5 | −871.5 | −871.4 | −871.6 | −871.0 | −871.0 | −871.0 | −871.0 | −839.4 | −840.5 | −841.5 | −839.2 | |

0.40 | −871.5 | −871.5 | −871.4 | −871.6 | −871.0 | −871.0 | −871.0 | −871.0 | −839.4 | −840.5 | −841.4 | −839.1 | |

0.45 | −871.5 | −871.4 | −871.3 | −871.5 | −871.0 | −871.0 | −871.0 | −870.9 | −839.3 | −840.5 | −841.4 | −839.1 | |

AIC | 0.50 | −871.5 | −871.4 | −871.3 | −871.5 | −870.9 | −871.0 | −871.0 | −870.9 | −839.3 | −840.4 | −841.4 | −839.1 |

0.55 | −871.4 | −871.4 | −871.3 | −871.5 | −870.9 | −870.9 | −870.9 | −870.9 | −839.3 | −840.4 | −841.4 | −839.1 | |

0.60 | −871.4 | −871.4 | −871.3 | −871.5 | −870.9 | −870.9 | −870.9 | −870.9 | −839.3 | −840.4 | −841.3 | −839.0 | |

0.65 | −871.4 | −871.4 | −871.3 | −871.4 | −870.9 | −870.9 | −870.9 | −870.8 | −839.2 | −840.4 | −841.3 | −839.0 | |

0.70 | −871.3 | −871.3 | −871.3 | −871.4 | −870.8 | −870.9 | −870.9 | −870.8 | −839.2 | −840.3 | −841.3 | −839.0 | |

0.75 | −871.3 | −871.3 | −871.2 | −871.4 | −870.8 | −870.8 | −870.9 | −870.8 | −839.2 | −840.3 | −841.2 | −838.9 | |

0.80 | −871.3 | −871.3 | −871.2 | −871.3 | −870.8 | −870.8 | −870.8 | −870.8 | −839.1 | −840.2 | −841.2 | −838.9 | |

0.85 | −871.2 | −871.3 | −871.2 | −871.3 | −870.7 | −870.8 | −870.8 | −870.7 | −839.1 | −840.2 | −841.1 | −838.8 | |

0.90 | −871.2 | −871.2 | −871.2 | −871.3 | −870.7 | −870.7 | −870.7 | −870.7 | −839.0 | −840.1 | −841.1 | −838.8 | |

0.95 | −871.2 | −871.2 | −871.1 | −871.2 | −870.6 | −870.7 | −870.7 | −870.6 | −839.9 | −840.6 | −840.9 | −839.7 | |

0.05 | −839.7 | −839.6 | −839.4 | −839.8 | −839.2 | −839.2 | −839.1 | −839.2 | −808.2 | −809.4 | −810.3 | −808.0 | |

0.10 | −839.7 | −839.5 | −839.4 | −839.7 | −839.1 | −839.1 | −839.1 | −839.1 | −807.8 | −808.9 | −809.8 | −807.6 | |

0.15 | −839.6 | −839.5 | −839.3 | −839.7 | −839.1 | −839.0 | −839.0 | −839.0 | −807.6 | −808.7 | −809.6 | −807.4 | |

0.20 | −839.5 | −839.4 | −839.3 | −839.6 | −839.0 | −839.0 | −839.0 | −839.0 | −807.5 | −808.6 | −809.5 | −807.2 | |

0.25 | −839.5 | −839.4 | −839.3 | −839.6 | −839.0 | −839.0 | −839.0 | −838.9 | −807.4 | −808.5 | −809.5 | −807.2 | |

0.30 | −839.5 | −839.4 | −839.3 | −839.5 | −838.9 | −838.9 | −838.9 | −838.9 | −807.3 | −808.5 | −809.4 | −807.1 | |

0.35 | −839.4 | −839.4 | −839.3 | −839.5 | −838.9 | −838.9 | −838.9 | −838.9 | −807.3 | −808.4 | −809.4 | −807.1 | |

0.40 | −839.4 | −839.3 | −839.2 | −839.5 | −838.9 | −838.9 | −838.9 | −838.9 | −807.2 | −808.4 | −809.3 | −807.0 | |

0.45 | −839.4 | −839.3 | −839.2 | −839.4 | −838.8 | −838.9 | −838.9 | −838.8 | −807.2 | −808.4 | −809.3 | −807.0 | |

BIC | 0.50 | −839.3 | −839.3 | −839.2 | −839.4 | −838.8 | −838.8 | −838.8 | −838.8 | −807.2 | −808.3 | −809.3 | −807.0 |

0.55 | −839.3 | −839.3 | −839.2 | −839.4 | −838.8 | −838.8 | −838.8 | −838.8 | −807.2 | −808.3 | −809.2 | −806.9 | |

0.60 | −839.3 | −839.3 | −839.2 | −839.3 | −838.8 | −838.8 | −838.8 | −838.7 | −807.1 | −808.3 | −809.2 | −806.9 | |

0.65 | −839.3 | −839.2 | −839.2 | −839.3 | −838.7 | −838.8 | −838.8 | −838.7 | −807.1 | −808.2 | −809.2 | −806.9 | |

0.70 | −839.2 | −839.2 | −839.1 | −839.3 | −838.7 | −838.7 | −838.8 | −838.7 | −807.1 | −808.2 | −809.2 | −806.8 | |

0.75 | −839.2 | −839.2 | −839.1 | −839.2 | −838.7 | −838.7 | −838.7 | −838.7 | −807.0 | −808.2 | −809.1 | −806.8 | |

0.80 | −839.2 | −839.2 | −839.1 | −839.2 | −838.7 | −838.7 | −838.7 | −838.6 | −807.0 | −808.1 | −809.1 | −806.8 | |

0.85 | −839.1 | −839.1 | −839.1 | −839.2 | −838.6 | −838.6 | −838.7 | −838.6 | −806.9 | −808.1 | −809.0 | −806.7 | |

0.90 | −839.1 | −839.1 | −839.0 | −839.1 | −838.6 | −838.6 | −838.6 | −838.6 | −806.9 | −808.0 | −808.9 | −806.7 | |

0.95 | −839.0 | −839.1 | −839.0 | −839.1 | −838.5 | −838.5 | −838.6 | −838.5 | −807.7 | −808.5 | −808.8 | −807.5 |

**Table A2.**AIC and BIC criteria for the RPGJSB2${}_{q}$ model of the COVID-19 data set considering 3 options for G (normal, logistic, and Cauchy) and the 4 discussed link functions.

Normal | Logistic | Cauchy | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Criteria | $\mathit{q}$ | Logit | Probit | Loglog | Cloglog | Logit | Probit | Loglog | Cloglog | Logit | Probit | Loglog | Cloglog |

0.05 | −869.0 | −872.3 | −873.4 | −868.7 | −857.4 | −863.8 | −867.6 | −856.9 | −758.5 | −770.5 | −779.3 | −757.6 | |

0.10 | −869.8 | −872.7 | −873.5 | −869.6 | −860.0 | −865.7 | −869.1 | −859.6 | −779.2 | −789.0 | −796.4 | −778.4 | |

0.15 | −870.4 | −872.9 | −873.5 | −870.1 | −862.1 | −867.2 | −870.1 | −861.7 | −795.1 | −803.3 | −809.4 | −794.5 | |

0.20 | −870.8 | −873.1 | −873.4 | −870.6 | −864.0 | −868.5 | −871.0 | −863.6 | −808.5 | −815.2 | −820.1 | −808.0 | |

0.25 | −871.2 | −873.2 | −873.4 | −871.0 | −865.7 | −869.7 | −871.8 | −865.3 | −819.6 | −825.0 | −828.9 | −819.1 | |

0.30 | −871.6 | −873.3 | −873.3 | −871.4 | −867.2 | −870.7 | −872.3 | −866.9 | −828.4 | −832.6 | −835.7 | −828.0 | |

0.35 | −871.9 | −873.4 | −873.1 | −871.7 | −868.7 | −871.6 | −872.8 | −868.4 | −835.0 | −838.1 | −840.4 | −834.6 | |

0.40 | −872.2 | −873.4 | −873.0 | −872.0 | −870.0 | −872.3 | −873.0 | −869.8 | −839.3 | −841.4 | −843.0 | −839.0 | |

0.45 | −872.4 | −873.4 | −872.8 | −872.3 | −871.2 | −872.7 | −872.9 | −871.0 | −841.2 | −842.5 | −843.3 | −841.0 | |

AIC | 0.50 | −872.6 | −873.4 | −872.5 | −872.5 | −872.1 | −872.9 | −872.6 | −872.0 | −840.6 | −841.0 | −841.1 | −840.4 |

0.55 | −872.9 | −873.3 | −872.2 | −872.8 | −872.7 | −872.8 | −871.8 | −872.7 | −837.1 | −836.6 | −836.0 | −837.0 | |

0.60 | −873.0 | −873.2 | −871.8 | −873.0 | −872.9 | −872.0 | −870.4 | −872.9 | −830.1 | −828.7 | −827.5 | −830.1 | |

0.65 | −873.2 | −873.0 | −871.3 | −873.2 | −872.4 | −870.5 | −868.1 | −872.4 | −818.9 | −816.7 | −814.8 | −818.9 | |

0.70 | −873.3 | −872.7 | −870.8 | −873.3 | −870.8 | −867.9 | −864.6 | −871.0 | −802.3 | −799.2 | −796.6 | −802.4 | |

0.75 | −873.3 | −872.3 | −870.0 | −873.4 | −867.7 | −863.5 | −859.3 | −868.0 | −778.5 | −774.5 | −771.2 | −778.6 | |

0.80 | −873.2 | −871.7 | −868.9 | −873.3 | −862.2 | −856.5 | −851.1 | −862.6 | −744.2 | −739.3 | −735.4 | −744.3 | |

0.85 | −873.0 | −870.8 | −867.5 | −873.1 | −852.8 | −845.3 | −838.5 | −853.3 | −693.1 | −687.5 | −682.9 | −693.4 | |

0.90 | −872.3 | −869.3 | −865.2 | −872.5 | −836.9 | −827.1 | −818.5 | −837.6 | −611.5 | −605.2 | −600.1 | −611.8 | |

0.95 | −870.7 | −866.3 | −861.1 | −871.1 | −810.5 | −797.4 | −786.0 | −811.6 | −453.7 | −446.8 | −431.4 | −454.1 | |

0.05 | −839.8 | −843.1 | −844.2 | −839.5 | −828.2 | −834.6 | −838.4 | −827.7 | −729.3 | −741.3 | −750.1 | −728.4 | |

0.10 | −840.6 | −843.5 | −844.3 | −840.4 | −830.8 | −836.5 | −839.9 | −830.4 | −750.0 | −759.8 | −767.2 | −749.2 | |

0.15 | −841.2 | −843.7 | −844.3 | −840.9 | −832.9 | −838.0 | −840.9 | −832.5 | −765.9 | −774.1 | −780.2 | −765.3 | |

0.20 | −841.6 | −843.9 | −844.2 | −841.4 | −834.8 | −839.3 | −841.8 | −834.4 | −779.3 | −786.0 | −790.9 | −778.8 | |

0.25 | −842.0 | −844.0 | −844.2 | −841.8 | −836.5 | −840.5 | −842.6 | −836.1 | −790.4 | −795.8 | −799.7 | −789.9 | |

0.30 | −842.4 | −844.1 | −844.1 | −842.2 | −838.0 | −841.5 | −843.1 | −837.7 | −799.2 | −803.4 | −806.5 | −798.8 | |

0.35 | −842.7 | −844.2 | −843.9 | −842.5 | −839.5 | −842.4 | −843.6 | −839.2 | −805.8 | −808.9 | −811.2 | −805.4 | |

0.40 | −843.0 | −844.2 | −843.8 | −842.8 | −840.8 | −843.1 | −843.8 | −840.6 | −810.1 | −812.2 | −813.8 | −809.8 | |

0.45 | −843.2 | −844.2 | −843.6 | −843.1 | −842.0 | −843.5 | −843.7 | −841.8 | −812.0 | −813.3 | −814.1 | −811.8 | |

BIC | 0.50 | −843.4 | −844.2 | −843.3 | −843.3 | −842.9 | −843.7 | −843.4 | −842.8 | −811.4 | −811.8 | −811.9 | −811.2 |

0.55 | −843.7 | −844.1 | −843.0 | −843.6 | −843.5 | −843.6 | −842.6 | −843.5 | −807.9 | −807.4 | −806.8 | −807.8 | |

0.60 | −843.8 | −844.0 | −842.6 | −843.8 | −843.7 | −842.8 | −841.2 | −843.7 | −800.9 | −799.5 | −798.3 | −800.9 | |

0.65 | −844.0 | −843.8 | −842.1 | −844.0 | −843.2 | −841.3 | −838.9 | −843.2 | −789.7 | −787.5 | −785.6 | −789.7 | |

0.70 | −844.1 | −843.5 | −841.6 | −844.1 | −841.6 | −838.7 | −835.4 | −841.8 | −773.1 | −770.0 | −767.4 | −773.2 | |

0.75 | −844.1 | −843.1 | −840.8 | −844.2 | −838.5 | −834.3 | −830.1 | −838.8 | −749.3 | −745.3 | −742.0 | −749.4 | |

0.80 | −844.0 | −842.5 | −839.7 | −844.1 | −833.0 | −827.3 | −821.9 | −833.4 | −715.0 | −710.1 | −706.2 | −715.1 | |

0.85 | −843.8 | −841.6 | −838.3 | −843.9 | −823.6 | −816.1 | −809.3 | −824.1 | −663.9 | −658.3 | −653.7 | −664.2 | |

0.90 | −843.1 | −840.1 | −836.0 | −843.3 | −807.7 | −797.9 | −789.3 | −808.4 | −582.3 | −576.0 | −570.9 | −582.6 | |

0.95 | −841.5 | −837.1 | −831.9 | −841.9 | −781.3 | −768.2 | −756.8 | −782.4 | −424.5 | −417.6 | −402.2 | −424.9 |

#### Appendix B.2. Estimated Parameters

**Table A3.**The estimated parameters for different quantiles in the RPGJSB2${}_{q}$ model of the COVID-19 data set, with the normal distribution for G and the loglog link. Furthermore, we present the p-values for the traditional normality test for the randomized quantile residuals.

p-Values for Quantile Residuals | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{q}$ | Parameter | Estimated | s.e. | t-Value | p-Value | KS | SW | AD | CVM |

0.1 | ${\beta}_{0}$ | −2.0810 | 0.1154 | −18.03 | <0.0001 | 0.441 | 0.260 | 0.128 | 0.099 |

${\beta}_{1}$ | 0.0150 | 0.0102 | 1.46 | 0.0716 | |||||

${\beta}_{2}$ | 0.0153 | 0.0077 | 1.99 | 0.0234 | |||||

${\beta}_{3}$ | 0.1310 | 0.0423 | 3.10 | 0.0010 | |||||

${\beta}_{4}$ | 0.0967 | 0.0371 | 2.61 | 0.0045 | |||||

${\nu}_{0}$ | 1.5341 | 0.0881 | 17.41 | <0.0001 | |||||

${\nu}_{1}$ | 0.1210 | 0.1608 | 0.75 | 0.2259 | |||||

${\nu}_{2}$ | 0.1801 | 0.1474 | 1.22 | 0.1110 | |||||

0.25 | ${\beta}_{0}$ | −1.9443 | 0.1140 | −17.06 | <0.0001 | 0.573 | 0.324 | 0.164 | 0.119 |

${\beta}_{1}$ | 0.0144 | 0.0103 | 1.40 | 0.0814 | |||||

${\beta}_{2}$ | 0.0156 | 0.0078 | 2.01 | 0.0223 | |||||

${\beta}_{3}$ | 0.1158 | 0.0331 | 3.50 | 0.0002 | |||||

${\beta}_{4}$ | 0.0756 | 0.0290 | 2.61 | 0.0046 | |||||

${\nu}_{0}$ | 1.7178 | 0.0873 | 19.69 | <0.0001 | |||||

${\nu}_{1}$ | 0.1263 | 0.1600 | 0.79 | 0.2149 | |||||

${\nu}_{2}$ | 0.1914 | 0.1462 | 1.31 | 0.0953 | |||||

0.75 | ${\beta}_{0}$ | −1.7334 | 0.1144 | −15.16 | <0.0001 | 0.396 | 0.070 | 0.048 | 0.035 |

${\beta}_{1}$ | 0.0139 | 0.0108 | 1.29 | 0.0989 | |||||

${\beta}_{2}$ | 0.0161 | 0.0082 | 1.97 | 0.0244 | |||||

${\beta}_{3}$ | 0.0909 | 0.0330 | 2.76 | 0.0029 | |||||

${\beta}_{4}$ | 0.0364 | 0.0289 | 1.26 | 0.1041 | |||||

${\nu}_{0}$ | 2.1731 | 0.0843 | 25.78 | <0.0001 | |||||

${\nu}_{1}$ | 0.1174 | 0.1564 | 0.75 | 0.2264 | |||||

${\nu}_{2}$ | 0.2170 | 0.1412 | 1.54 | 0.0622 | |||||

0.9 | ${\beta}_{0}$ | -1.6473 | 0.1163 | −14.16 | <0.0001 | 0.169 | 0.005 | 0.006 | 0.005 |

${\beta}_{1}$ | 0.0142 | 0.0113 | 1.26 | 0.1044 | |||||

${\beta}_{2}$ | 0.0161 | 0.0086 | 1.87 | 0.0304 | |||||

${\beta}_{3}$ | 0.0860 | 0.0367 | 2.34 | 0.0096 | |||||

${\beta}_{4}$ | 0.0162 | 0.0311 | 0.52 | 0.3018 | |||||

${\nu}_{0}$ | 2.5206 | 0.0818 | 30.81 | <0.0001 | |||||

${\nu}_{1}$ | 0.0877 | 0.1534 | 0.57 | 0.2837 | |||||

${\nu}_{2}$ | 0.2296 | 0.1363 | 1.68 | 0.0460 |

#### Appendix B.3. Additional Information for Local Influence

**Figure A1.**The index plots of ${C}_{i}$ for $\widehat{\mathsf{\beta}}$ (

**upper**) and $\widehat{\mathsf{\nu}}$ (

**lower**) under the weight perturbation (

**left**), response perturbation (

**center**), and covariate perturbation (

**right**) schemes for the RPGJSB2${}_{q}$ model with $q=0.1$ (link loglog and G the cdf from the normal model) of the COVID-19 data set.

**Figure A2.**The index plots of ${C}_{i}$ for $\widehat{\mathsf{\beta}}$ (

**upper**) and $\widehat{\mathsf{\nu}}$ (

**lower**) under the weight perturbation (

**left**), response perturbation (

**center**), and covariate perturbation (

**right**) schemes for the RPGJSB2${}_{q}$ model with $q=0.25$ (link loglog and G the cdf from the normal model) of the COVID-19 data set.

**Figure A3.**The index plots of ${C}_{i}$ for $\widehat{\mathsf{\beta}}$ (

**upper**) and $\widehat{\mathsf{\nu}}$ (

**lower**) under the weight perturbation (

**left**), response perturbation (

**center**), and covariate perturbation (

**right**) schemes for the RPGJSB2${}_{q}$ model with $q=0.75$ (link loglog and G the cdf from the normal model) of the COVID-19 data set.

**Figure A4.**The index plots of ${C}_{i}$ for $\widehat{\mathsf{\beta}}$ (

**upper**) and $\widehat{\mathsf{\nu}}$ (

**lower**) under the weight perturbation (

**left**), response perturbation (

**center**), and covariate perturbation (

**right**) schemes for the RPGJSB2${}_{q}$ model with $q=0.9$ (link loglog and G the cdf from the normal model) of the COVID-19 data set.

## References

- Ferrari, S.L.; Cribari-Neto, F. Beta regression for modelling rates and proportions. J. Appl. Stat.
**2004**, 31, 799–815. [Google Scholar] [CrossRef] - Ospina, R.; Ferrari, S.L.P. Inflated beta distributions. Stat. Pap.
**2008**, 51, 111–126. [Google Scholar] [CrossRef] [Green Version] - Bayes, C.L.; Bazán, J.L.; García, C. A new robust regression model for proportions. Bayesian Anal.
**2012**, 7, 841–866. [Google Scholar] [CrossRef] - Migliorati, S.; Di Brisco, A.M.; Ongaro, A. A New Regression Model for Bounded Responses. Bayesian Anal.
**2018**, 13, 845–872. [Google Scholar] [CrossRef] - Koenker, R.; Bassett, G. Regression quantiles. Econometrica
**1978**, 46, 33–50. [Google Scholar] [CrossRef] - Lemonte, A.J.; Moreno-Arenas, G. On a heavy-tailed parametric quantile regression model for limited range response variables. Comput. Stat.
**2020**, 35, 379–398. [Google Scholar] [CrossRef] - Su, S. Flexible parametric quantile regression model. Stat. Comput.
**2015**, 25, 635–650. [Google Scholar] [CrossRef] - Mazucheli, J.; Menezes, A.F.B.; Fernandes, L.B.; Oliveira, R.P.; Ghitany, M.E. The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modelling of quantiles conditional on covariates. J. Appl. Stat.
**2020**, 47, 954–974. [Google Scholar] [CrossRef] - Lemonte, A.J.; Bazán, J.L. New class of Johnson S
_{B}distributions and its associated regression model for rates and proportions. Biom. J.**2016**, 58, 727–746. [Google Scholar] [CrossRef] [PubMed] - Johnson, N.L. Systems of frequency curves generated by the methods of translation. Biometrika
**1949**, 36, 149–176. [Google Scholar] [CrossRef] - Cancho, V.G.; Bazán, J.L.; Dey, D.K. A new class of regression model for a bounded response with application in the study of the incidence rate of colorectal cancer. Stat. Methods Med. Res.
**2020**, 29, 2015–2033. [Google Scholar] [CrossRef] [PubMed] - Bayes, C.L.; Bazán, J.L.; De Castro, M. A quantile parametric mixed regression model for bounded response variables. Stat. Interface
**2017**, 10, 483–493. [Google Scholar] [CrossRef] - Durrans, S.R. Distributions of fractional order statistics in hydrology. Water Resour. Res.
**1992**, 28, 1649–1655. [Google Scholar] [CrossRef] - Lehmann, E.L. The power of rank tests. Ann. Math. Stat.
**1953**, 24, 23–43. [Google Scholar] [CrossRef] - Fletcher, R. Practical Methods of Optimization, 2nd ed.; John Wiley & Sons.: New York, NY, USA, 1987. [Google Scholar]
- Cox, D.; Hinkley, D. Theoretical Statistics; Chapman and Hall: London, UK, 1974. [Google Scholar]
- Dunn, P.K.; Smyth, G.K. Randomized quantile residuals. J. Comput. Graph. Stat.
**1996**, 5, 236–244. [Google Scholar] - Yap, B.W.; Sim, C.H. Comparisons of various normality tests. J. Stat. Comput. Simul.
**2011**, 81, 2141–2155. [Google Scholar] [CrossRef] - Cook, R.D. Assessment of Local Influence. J. R. Soc. B Stat. Methodol.
**1986**, 48, 133–155. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2021; Available online: https://www.R-project.org/ (accessed on 10 January 2022).
- Du, R.H.; Liang, L.R.; Yang, C.Q.; Wang, W.; Cao, T.Z.; Li, M.; Guo, G.Y.; Du, J.; Zheng, C.L.; Zhu, Q.; et al. Predictors of mortality for patients with COVID-19 pneumonia caused by SARS-CoV-2: A prospective cohort study. Eur. Respir. J.
**2020**, 55, 2000524. [Google Scholar] [CrossRef] [Green Version] - Ji, J.S.; Liu, Y.; Liu, R.; Zha, Y.; Chang, X.; Zhang, L.; Zhang, Y.; Zeng, J.; Dong, T.; Xu, X.; et al. Survival analysis of hospital length of stay of novel coronavirus (COVID-19) pneumonia patients in Sichuan, China. medRxiv
**2020**. [Google Scholar] [CrossRef] - Li, X.; Xu, S.; Yu, M.; Wang, K.; Tao, Y.; Zhou, Y.; Shi, J.; Zhou, M.; Wu, B.; Yang, Z. Risk factors for severity and mortality in adult COVID-19 inpatients in Wuhan. J. Allergy Clin. Immunol.
**2020**, 146, 110–118. [Google Scholar] [CrossRef] - Livingston, E.; Bucher, K. Coronavirus disease 2019 (COVID-19) in Italy. J. Am. Med Assoc.
**2020**, 323, 1335. [Google Scholar] [CrossRef] [PubMed] [Green Version] - WHO. Coronavirus Disease (COVID-19) Dashboard; World Health Organization: Geneva, Switzerland, 2021; Available online: https://covid19.who.int/ (accessed on 25 May 2021).

**Figure 1.**Pdf for RPGJSB1${}_{q}(\psi ,\delta =1,\alpha )$ model with logit link and $G=\Phi $.

**Left**panel: $q=0.25$, $\alpha =0.5$, and varying $\psi $;

**center**panel: $q=0.5$, $\alpha =0.5$, and varying $\psi $;

**right**panel: $q=0.5$, $\psi =0.4$, and varying $\alpha $.

**Figure 2.**Descriptive plots for Q(

`mort`) versus log(

`surface`) and versus log(

`population`) for different link functions: logit, probit, loglog, and cloglog and separated by continent: Africa, Asia, or Oceania (black), the Americas (red), and Europe (green).

**Figure 3.**Point estimation and 90%, 95%, and 99% confidence intervals for parameters estimated in the RPGJSB2${}_{q}$ model for different quantiles (loglog link and G the cdf of the normal model).

**Figure 4.**The estimated $100\times q$th quantile in the RPGJSB2${}_{q}$ model for $log\left(\mathtt{population}\right)=16$ (around 9 million people) varying the $log\left(\mathtt{surface}\right)$ for countries in Africa, Asia, or Oceania (

**left**panel), the Americas (

**center**panel), and Europe (

**right**panel) considering the cloglog link and G the cdf from the logistic model.

**Figure 5.**The index plots of ${C}_{i}$ for $\widehat{\mathsf{\beta}}$ (

**upper**) and $\widehat{\mathsf{\nu}}$ (

**lower**) under the weight perturbation (

**left**), response perturbation (

**center**), and covariate perturbation (

**right**) schemes for the RPGJSB2${}_{q=0.5}$ model (cloglog link and G the cdf from the logistic model) of the COVID-19 data set.

Logistic | Normal | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Link | $\mathit{q}$ | ${\mathsf{\beta}}_{\mathbf{0}}$ | ${\mathsf{\beta}}_{\mathbf{1}}$ | ${\mathsf{\nu}}_{\mathbf{0}}$ | ${\mathsf{\nu}}_{\mathbf{1}}$ | $log\left(\mathsf{\alpha}\right)$ | ${\mathsf{\beta}}_{\mathbf{0}}$ | ${\mathsf{\beta}}_{\mathbf{1}}$ | ${\mathsf{\nu}}_{\mathbf{0}}$ | ${\mathsf{\nu}}_{\mathbf{1}}$ | $log\left(\mathsf{\alpha}\right)$ |

logit | 0.1 | 4.9 | 2.6 | 2.2 | 0.4 | −0.7 | 4.4 | 2.4 | 1.5 | 0.3 | −1.4 |

0.5 | 4.8 | 2.1 | 2.2 | 0.4 | −0.7 | 4.6 | 2.1 | 1.5 | 0.3 | −1.4 | |

0.9 | 4.7 | 1.8 | 2.2 | 0.4 | −0.7 | 4.8 | 1.9 | 1.5 | 0.3 | −1.4 | |

loglog | 0.1 | 1.3 | 0.8 | 0.8 | −0.3 | 0.1 | 1.2 | 0.7 | −0.1 | −0.3 | 1.1 |

0.5 | 2.1 | 0.9 | 1.0 | −0.2 | 0.1 | 2.0 | 0.9 | 0.0 | −0.3 | 1.0 | |

0.9 | 2.8 | 1.0 | 1.1 | −0.2 | 0.1 | 2.8 | 1.0 | 0.1 | −0.2 | 1.0 |

**Table 2.**Estimated bias, standard error of the estimated parameters ($S{E}_{1}$), the mean of the estimated standard errors ($S{E}_{2}$), and 95% coverage probabilities (CP), when G and the link are correctly specified (case G is the cdf of the logistic distribution).

$\mathit{n}=100$ | $\mathit{n}=200$ | $\mathit{n}=500$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{G}$ | Link | $\mathit{q}$ | Parameter | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP |

logistic | logit | 0.1 | ${\beta}_{0}$ | −0.034 | 0.753 | 0.728 | 0.938 | −0.017 | 0.538 | 0.529 | 0.946 | −0.007 | 0.345 | 0.339 | 0.946 |

${\beta}_{1}$ | −0.015 | 0.238 | 0.229 | 0.934 | −0.007 | 0.166 | 0.163 | 0.942 | −0.003 | 0.104 | 0.102 | 0.947 | |||

${\nu}_{0}$ | 0.041 | 0.381 | 0.367 | 0.935 | 0.020 | 0.269 | 0.263 | 0.942 | 0.009 | 0.171 | 0.170 | 0.947 | |||

${\nu}_{1}$ | −0.001 | 0.088 | 0.085 | 0.939 | 0.000 | 0.061 | 0.060 | 0.946 | 0.000 | 0.039 | 0.038 | 0.946 | |||

$log\left(\alpha \right)$ | −0.004 | 0.355 | 0.331 | 0.947 | −0.002 | 0.232 | 0.224 | 0.946 | −0.002 | 0.140 | 0.138 | 0.948 | |||

0.5 | ${\beta}_{0}$ | −0.017 | 0.485 | 0.472 | 0.941 | 0.001 | 0.322 | 0.319 | 0.946 | −0.003 | 0.204 | 0.205 | 0.950 | ||

${\beta}_{1}$ | −0.005 | 0.146 | 0.142 | 0.939 | 0.000 | 0.096 | 0.095 | 0.946 | −0.001 | 0.061 | 0.061 | 0.949 | |||

${\nu}_{0}$ | 0.046 | 0.452 | 0.443 | 0.946 | 0.027 | 0.296 | 0.294 | 0.948 | 0.007 | 0.183 | 0.182 | 0.949 | |||

${\nu}_{1}$ | 0.002 | 0.107 | 0.106 | 0.946 | 0.002 | 0.068 | 0.068 | 0.949 | 0.000 | 0.042 | 0.042 | 0.951 | |||

$log\left(\alpha \right)$ | 0.004 | 0.352 | 0.331 | 0.952 | −0.001 | 0.231 | 0.224 | 0.948 | 0.000 | 0.142 | 0.139 | 0.947 | |||

0.9 | ${\beta}_{0}$ | −0.001 | 0.620 | 0.591 | 0.930 | −0.002 | 0.369 | 0.363 | 0.942 | 0.002 | 0.237 | 0.236 | 0.950 | ||

${\beta}_{1}$ | 0.004 | 0.177 | 0.169 | 0.932 | 0.002 | 0.112 | 0.111 | 0.943 | 0.001 | 0.072 | 0.072 | 0.948 | |||

${\nu}_{0}$ | 0.060 | 0.461 | 0.443 | 0.943 | 0.024 | 0.289 | 0.283 | 0.943 | 0.007 | 0.184 | 0.182 | 0.946 | |||

${\nu}_{1}$ | 0.006 | 0.103 | 0.100 | 0.938 | 0.002 | 0.066 | 0.065 | 0.946 | 0.000 | 0.043 | 0.043 | 0.945 | |||

$log\left(\alpha \right)$ | 0.010 | 0.362 | 0.334 | 0.946 | 0.003 | 0.234 | 0.224 | 0.949 | 0.002 | 0.140 | 0.139 | 0.949 | |||

loglog | 0.1 | ${\beta}_{0}$ | 0.008 | 0.175 | 0.168 | 0.931 | 0.002 | 0.116 | 0.113 | 0.938 | 0.000 | 0.071 | 0.071 | 0.949 | |

${\beta}_{1}$ | 0.001 | 0.039 | 0.037 | 0.935 | 0.000 | 0.026 | 0.025 | 0.937 | 0.000 | 0.016 | 0.016 | 0.948 | |||

${\nu}_{0}$ | 0.020 | 0.413 | 0.398 | 0.944 | 0.005 | 0.280 | 0.275 | 0.946 | −0.001 | 0.165 | 0.167 | 0.950 | |||

${\nu}_{1}$ | 0.000 | 0.096 | 0.092 | 0.939 | −0.002 | 0.067 | 0.065 | 0.942 | −0.001 | 0.039 | 0.039 | 0.949 | |||

$log\left(\alpha \right)$ | 0.153 | 1.175 | 2.515 | 0.964 | 0.035 | 0.349 | 0.324 | 0.961 | 0.014 | 0.178 | 0.174 | 0.956 | |||

0.5 | ${\beta}_{0}$ | −0.002 | 0.130 | 0.128 | 0.944 | −0.003 | 0.090 | 0.090 | 0.951 | 0.001 | 0.061 | 0.061 | 0.946 | ||

${\beta}_{1}$ | 0.000 | 0.031 | 0.030 | 0.945 | −0.001 | 0.021 | 0.021 | 0.949 | 0.000 | 0.014 | 0.014 | 0.947 | |||

${\nu}_{0}$ | 0.007 | 0.386 | 0.376 | 0.944 | 0.003 | 0.264 | 0.261 | 0.947 | 0.006 | 0.175 | 0.175 | 0.950 | |||

${\nu}_{1}$ | −0.003 | 0.093 | 0.091 | 0.945 | −0.002 | 0.063 | 0.062 | 0.949 | 0.000 | 0.041 | 0.041 | 0.950 | |||

$log\left(\alpha \right)$ | 0.143 | 1.070 | 2.042 | 0.965 | 0.041 | 0.306 | 0.290 | 0.962 | 0.012 | 0.177 | 0.174 | 0.951 | |||

0.9 | ${\beta}_{0}$ | −0.005 | 0.178 | 0.175 | 0.939 | −0.004 | 0.141 | 0.139 | 0.942 | −0.002 | 0.082 | 0.082 | 0.947 | ||

${\beta}_{1}$ | −0.001 | 0.042 | 0.041 | 0.940 | −0.001 | 0.033 | 0.032 | 0.943 | 0.000 | 0.019 | 0.019 | 0.947 | |||

${\nu}_{0}$ | 0.012 | 0.387 | 0.374 | 0.940 | 0.010 | 0.296 | 0.288 | 0.944 | 0.004 | 0.174 | 0.173 | 0.949 | |||

${\nu}_{1}$ | −0.002 | 0.094 | 0.091 | 0.940 | 0.000 | 0.071 | 0.069 | 0.944 | 0.000 | 0.041 | 0.041 | 0.949 | |||

$log\left(\alpha \right)$ | 0.133 | 0.968 | 1.596 | 0.965 | 0.042 | 0.311 | 0.290 | 0.961 | 0.014 | 0.177 | 0.174 | 0.952 |

**Table 3.**Estimated bias, standard error of the estimated parameters ($S{E}_{1}$), the mean of the estimated standard errors ($S{E}_{2}$), and 95% coverage probabilities (CP), when G and the link are correctly specified (case G is the cdf of the normal distribution).

$\mathit{n}=100$ | $\mathit{n}=200$ | $\mathit{n}=500$ | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{G}$ | Link | $\mathit{q}$ | Parameter | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP | Bias | ${\mathit{S}\mathit{E}}_{\mathbf{1}}$ | ${\mathit{S}\mathit{E}}_{\mathbf{2}}$ | CP |

normal | logit | 0.1 | ${\beta}_{0}$ | −0.004 | 0.725 | 0.711 | 0.939 | 0.000 | 0.470 | 0.473 | 0.952 | −0.002 | 0.289 | 0.291 | 0.951 |

${\beta}_{1}$ | −0.005 | 0.202 | 0.198 | 0.939 | −0.002 | 0.133 | 0.134 | 0.951 | −0.002 | 0.081 | 0.082 | 0.949 | |||

${\nu}_{0}$ | 0.912 | 2.171 | 0.730 | 0.847 | 0.243 | 1.024 | 0.450 | 0.946 | 0.045 | 0.270 | 0.248 | 0.954 | |||

${\nu}_{1}$ | 0.000 | 0.083 | 0.079 | 0.932 | 0.001 | 0.055 | 0.054 | 0.945 | 0.000 | 0.032 | 0.032 | 0.951 | |||

$log\left(\alpha \right)$ | −1.763 | 4.569 | 1.650 | 0.867 | −0.462 | 2.167 | 1.019 | 0.960 | −0.082 | 0.612 | 0.565 | 0.956 | |||

0.5 | ${\beta}_{0}$ | −0.006 | 0.464 | 0.452 | 0.942 | −0.005 | 0.330 | 0.324 | 0.945 | 0.002 | 0.196 | 0.194 | 0.946 | ||

${\beta}_{1}$ | −0.004 | 0.135 | 0.131 | 0.940 | −0.002 | 0.094 | 0.092 | 0.941 | 0.000 | 0.056 | 0.056 | 0.946 | |||

${\nu}_{0}$ | 0.944 | 2.251 | 0.703 | 0.841 | 0.215 | 0.966 | 0.450 | 0.949 | 0.040 | 0.281 | 0.250 | 0.952 | |||

${\nu}_{1}$ | 0.002 | 0.082 | 0.079 | 0.939 | 0.001 | 0.056 | 0.055 | 0.947 | 0.000 | 0.034 | 0.033 | 0.950 | |||

$log\left(\alpha \right)$ | −1.806 | 4.729 | 1.597 | 0.862 | −0.398 | 2.046 | 1.012 | 0.961 | −0.071 | 0.625 | 0.564 | 0.954 | |||

0.9 | ${\beta}_{0}$ | −0.028 | 0.595 | 0.550 | 0.910 | −0.001 | 0.393 | 0.375 | 0.934 | −0.004 | 0.244 | 0.242 | 0.947 | ||

${\beta}_{1}$ | −0.002 | 0.165 | 0.153 | 0.912 | 0.003 | 0.111 | 0.106 | 0.933 | 0.000 | 0.069 | 0.069 | 0.949 | |||

${\nu}_{0}$ | 0.923 | 2.248 | 0.712 | 0.852 | 0.235 | 1.009 | 0.450 | 0.947 | 0.047 | 0.279 | 0.253 | 0.949 | |||

${\nu}_{1}$ | 0.006 | 0.088 | 0.084 | 0.937 | 0.001 | 0.057 | 0.055 | 0.941 | 0.001 | 0.035 | 0.035 | 0.949 | |||

$log\left(\alpha \right)$ | −1.733 | 4.706 | 1.576 | 0.871 | −0.434 | 2.133 | 1.008 | 0.961 | −0.083 | 0.614 | 0.563 | 0.956 | |||

loglog | 0.1 | ${\beta}_{0}$ | 0.005 | 0.156 | 0.152 | 0.935 | 0.005 | 0.115 | 0.114 | 0.942 | 0.001 | 0.070 | 0.069 | 0.946 | |

${\beta}_{1}$ | 0.000 | 0.035 | 0.034 | 0.936 | 0.001 | 0.026 | 0.026 | 0.945 | 0.000 | 0.016 | 0.015 | 0.946 | |||

${\nu}_{0}$ | 0.085 | 0.834 | 0.530 | 0.963 | 0.024 | 0.371 | 0.351 | 0.951 | 0.006 | 0.209 | 0.209 | 0.952 | |||

${\nu}_{1}$ | −0.004 | 0.077 | 0.076 | 0.942 | −0.001 | 0.059 | 0.058 | 0.946 | −0.001 | 0.035 | 0.035 | 0.951 | |||

$log\left(\alpha \right)$ | 1.090 | 23.978 | 3.284 | 0.965 | 0.103 | 1.677 | 1.200 | 0.958 | 0.027 | 0.661 | 0.658 | 0.961 | |||

0.5 | ${\beta}_{0}$ | 0.002 | 0.116 | 0.114 | 0.942 | 0.000 | 0.084 | 0.083 | 0.946 | 0.000 | 0.054 | 0.053 | 0.948 | ||

${\beta}_{1}$ | 0.000 | 0.026 | 0.025 | 0.942 | 0.000 | 0.019 | 0.019 | 0.947 | 0.000 | 0.012 | 0.012 | 0.947 | |||

${\nu}_{0}$ | 0.123 | 0.990 | 0.539 | 0.954 | 0.017 | 0.379 | 0.348 | 0.955 | 0.009 | 0.212 | 0.212 | 0.952 | |||

${\nu}_{1}$ | −0.004 | 0.082 | 0.079 | 0.939 | −0.002 | 0.059 | 0.059 | 0.946 | −0.001 | 0.036 | 0.036 | 0.950 | |||

$log\left(\alpha \right)$ | 0.612 | 16.917 | 3.114 | 0.964 | 0.091 | 1.453 | 1.150 | 0.963 | 0.017 | 0.654 | 0.645 | 0.957 | |||

0.9 | ${\beta}_{0}$ | −0.016 | 0.224 | 0.219 | 0.935 | −0.007 | 0.169 | 0.167 | 0.939 | −0.004 | 0.095 | 0.095 | 0.943 | ||

${\beta}_{1}$ | −0.002 | 0.051 | 0.050 | 0.937 | −0.001 | 0.040 | 0.039 | 0.940 | −0.001 | 0.022 | 0.022 | 0.946 | |||

${\nu}_{0}$ | 0.125 | 0.934 | 0.538 | 0.958 | 0.029 | 0.386 | 0.357 | 0.951 | 0.008 | 0.208 | 0.207 | 0.951 | |||

${\nu}_{1}$ | 0.000 | 0.083 | 0.080 | 0.940 | 0.000 | 0.061 | 0.060 | 0.948 | 0.000 | 0.035 | 0.034 | 0.952 | |||

$log\left(\alpha \right)$ | 0.428 | 12.877 | 2.696 | 0.964 | 0.088 | 1.443 | 1.192 | 0.957 | 0.034 | 0.657 | 0.647 | 0.961 |

**Table 4.**Percentage of time where the maximization algorithm converges with theinitial value as the vector zero.

$\mathit{q}=0.1$ | $\mathit{q}=0.5$ | $\mathit{q}=0.9$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

$\mathit{G}$ | Link | 100 | 200 | 500 | 100 | 200 | 500 | 100 | 200 | 500 |

logistic | logit | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

loglog | 99.71 | 100.00 | 100.00 | 99.83 | 100.00 | 100.00 | 99.85 | 100.00 | 100.00 | |

normal | logit | 90.77 | 98.40 | 100.00 | 89.43 | 98.65 | 99.99 | 90.38 | 98.59 | 99.99 |

loglog | 99.43 | 99.99 | 100.00 | 99.01 | 99.98 | 100.00 | 99.05 | 99.98 | 100.00 |

**Table 5.**The estimated parameters and standard errors (s.e.) for different quantiles in the RPGJSB2${}_{q=0.5}$ model of the COVID-19 data set with G the cdf of the normal model and loglog link. The p-values for the traditional normality test for the RQRs are also presented.

p-Values for Quantile Residuals | |||||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{q}$ | Parameter | Estimated | s.e. | $\mathit{z}$-Value | $\mathit{p}$-Value | KS | SW | AD | CVM |

0.5 | ${\beta}_{0}$ | −1.8396 | 0.1136 | −16.19 | <0.0001 | 0.626 | 0.249 | 0.133 | 0.094 |

${\beta}_{1}$ | 0.0140 | 0.0105 | 1.34 | 0.0899 | |||||

${\beta}_{2}$ | 0.0159 | 0.0079 | 2.01 | 0.0223 | |||||

${\beta}_{3}$ | 0.1032 | 0.0308 | 3.35 | 0.0004 | |||||

${\beta}_{4}$ | 0.0575 | 0.0272 | 2.12 | 0.0171 | |||||

${\nu}_{0}$ | 1.9051 | 0.0862 | 22.11 | <0.0001 | |||||

${\nu}_{1}$ | 0.1268 | 0.1587 | 0.80 | 0.2121 | |||||

${\nu}_{2}$ | 0.2028 | 0.1445 | 1.40 | 0.0802 |

**Table 6.**The relative changes (in %) in the ML estimates (RC) and their corresponding standard errors (RCSE) for the indicated parameter and respective p-values for the COVID-19 data set, when observation 100 is dropped.

q | ||||||
---|---|---|---|---|---|---|

Parameter | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 | |

RC | 239.95 | 243.30 | 253.85 | 267.69 | 279.66 | |

RCSE | ${\beta}_{0}\left(q\right)$ | 18.10 | 18.71 | 19.62 | 20.63 | 21.45 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | |

RC | 298.10 | 295.33 | 296.05 | 292.45 | 276.05 | |

RCSE | ${\beta}_{1}\left(q\right)$ | 224.15 | 228.55 | 233.01 | 233.17 | 223.76 |

p-value | 0.1751 | 0.1879 | 0.1979 | 0.2067 | 0.2151 | |

RC | 313.98 | 310.46 | 305.38 | 300.83 | 300.56 | |

RCSE | ${\beta}_{2}\left(q\right)$ | 161.89 | 158.80 | 155.11 | 150.94 | 147.41 |

p-value | 0.0508 | 0.0491 | 0.0479 | 0.0469 | 0.046 | |

RC | 398.39 | 402.09 | 431.38 | 472.57 | 475.72 | |

RCSE | ${\beta}_{3}\left(q\right)$ | 104.18 | 103.37 | 108.53 | 113.21 | 109.20 |

p-value | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0002 | |

RC | 406.56 | 446.90 | 569.98 | 895.37 | 2014.00 | |

RCSE | ${\beta}_{4}\left(q\right)$ | 139.08 | 150.33 | 184.91 | 270.23 | 566.60 |

p-value | 0.0041 | 0.0037 | 0.0039 | 0.0044 | 0.0054 | |

RC | 96.05 | 92.62 | 90.77 | 89.93 | 89.73 | |

RCSE | ${\nu}_{0}\left(q\right)$ | 0.02 | 0.01 | 0.06 | 0.13 | 0.21 |

p-value | 0.4717 | 0.1467 | 0.0439 | 0.0120 | 0.0029 | |

RC | 112.14 | 113.02 | 116.81 | 138.60 | 224.79 | |

RCSE | ${\nu}_{1}\left(q\right)$ | 0.41 | 0.90 | 1.91 | 3.98 | 8.62 |

p-value | 0.1048 | 0.0950 | 0.0879 | 0.0820 | 0.0767 | |

RC | 15.21 | 11.87 | 6.52 | 0.32 | 4.46 | |

RCSE | ${\nu}_{2}\left(q\right)$ | 0.18 | 0.17 | 0.92 | 2.27 | 4.21 |

p-value | 0.1493 | 0.1440 | 0.1399 | 0.1362 | 0.1329 |

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## Share and Cite

**MDPI and ACS Style**

Gallardo, D.I.; Bourguignon, M.; Gómez, Y.M.; Caamaño-Carrillo, C.; Venegas, O.
Parametric Quantile Regression Models for Fitting Double Bounded Response with Application to COVID-19 Mortality Rate Data. *Mathematics* **2022**, *10*, 2249.
https://doi.org/10.3390/math10132249

**AMA Style**

Gallardo DI, Bourguignon M, Gómez YM, Caamaño-Carrillo C, Venegas O.
Parametric Quantile Regression Models for Fitting Double Bounded Response with Application to COVID-19 Mortality Rate Data. *Mathematics*. 2022; 10(13):2249.
https://doi.org/10.3390/math10132249

**Chicago/Turabian Style**

Gallardo, Diego I., Marcelo Bourguignon, Yolanda M. Gómez, Christian Caamaño-Carrillo, and Osvaldo Venegas.
2022. "Parametric Quantile Regression Models for Fitting Double Bounded Response with Application to COVID-19 Mortality Rate Data" *Mathematics* 10, no. 13: 2249.
https://doi.org/10.3390/math10132249