# Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications

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## Abstract

**:**

`R`package developed by the authors, named

`vasicekreg`, makes available the results of the present investigation. Applications with two real data sets are conducted for illustrative purposes: in one of them, the unit Vasicek quantile regression outperforms the models based on the Johnson-SB, Kumaraswamy, unit-logistic, and unit-Weibull distributions, whereas in the second one, the unit Vasicek mean regression outperforms the fits obtained by the beta and simplex distributions. Our investigation suggests that unit Vasicek quantile and mean regressions can be of practical usage as alternatives to some well-known models for analyzing data on the unit interval.

## 1. Introduction

## 2. The Vasicek Distribution

## 3. Maximum Likelihood Estimation and Residuals Analysis

`optim`function of

`R`[28]. As an alternative, to model $(\alpha ,\theta )$ or $(\mu ,\theta )$, conditional on covariates, we create two generalized additive model for location, scale, and shape (GAMLSS) frameworks that can be used directly in the

`gamlss`function of the

`gamlss`package [29,30,31] of

`R`. An advantage of having a GAMLSS structure is that all the parameters of the distribution can be modeled as linear, nonlinear, or smooth functions of covariates. In addition, we have available all the resources for the statistical modeling process within the

`gamlss`package of

`R`as model selection and diagnostics.

`gamlss`package, we may utilize the fitted normalized quantile residual [36] defined as

`gamlss`object, we can use these residuals to construct theoretical quantile versus empirical quantile (QQ) plots with simulated envelopes [38]. The main advantage of this simulation technique is its ease of interpretation without imposing any assumption on the residual distribution [39].

## 4. Simulation Studies

_{95%}) using the asymptotic normality of such ML estimators.

## 5. Applications

#### 5.1. The VASI Quantile Regression Model

^{2}) of the individuals, while the categorical covariates are related to gender (female or male) and ipaq (sedentary (S), insufficiently active (I), or active (A)). Observe that the ipaq is a questionnaire that permits the estimation of weekly time spent on physical activities of moderate and strong intensity, in different aspects of daily life, such as transportation, leisure, housework, and work, as well as the time spent in passive activities [42].

^{2}with a standard deviation of 3.15 kg/m

^{2}. The ipaq questionnaire classified individuals as follows: 76 individuals as insufficiently active, 60 sedentary, and 162 active. According to [41], the data set has one outlier, which consists of the individual #158 with the following characteristics: female, 49 years old, with BMI = 29.3 kg/m

^{2}and fat proportion in the arms equal to 0.196; in other words, this patient has a high BMI but low fat proportion in the arms.

- ${x}_{i1}$ is the (age
_{i}− 46.00) with 46.00 being the average age; - ${x}_{i2}$ is the (BMI
_{i}− 24.72) with 24.72 being the average BMI; - ${x}_{i3}$ is an indicator covariate, in which ${x}_{i3}=0$ for female or ${x}_{i3}=1$ for male;
- ${x}_{i4}$ is an indicator covariate, in which ${x}_{i4}=0$ for ipaq = S or ${x}_{i4}=1$ for ipaq = I;
- ${x}_{i5}$ is an indicator covariate, in which ${x}_{i5}=0$ for ipaq = S or ${x}_{i5}=1$ for ipaq = A.

^{2}, and ipaq = S. The parameter estimates for ${\delta}_{1}$ and ${\delta}_{2}$ indicate that the arm fat proportion is larger for older individuals with larger BMI. In contrast, the estimates for ${\delta}_{3}$, ${\delta}_{4}$, and ${\delta}_{5}$ have a negative influence on the arm fat proportion, indicating that this proportion is smaller for insufficiently active and active men, respectively. Consequently, the fat proportion is larger for women and sedentary individuals. As expected, $\widehat{\theta}$ does not vary with the quantiles, since it does not depend on covariates. From the variation in the parameters for different values of $\tau $, we can conclude that the estimated arm fat proportion depends on the quantiles. This conditional quantile variation rate, expressed by the estimated regression coefficients, is illustrated in Figure 3. We can also see that, from a statistical point of view, all covariates are significant at 5%, since they do not contemplate the value of zero in their respective confidence intervals.

`vasicekreg`package by means of the following

`R`codes:

`library(gamlss)`

`library(vasicekreg)`

`data(bodyfat, package = “vasicekreg”)`

`fit <- lapply(c(0.10, 0.25, 0.50, 0.75, 0.90), function(Tau)`

`{`

`tau <<- Tau;`

`gamlss(arms ~ age + sex2 + ipaq1 + ipaq2, data = bodyfat, trace = FALSE,`

`family = VASIQ(mu.link = “logit”, sigma.link = “logit”))`

`})`

`sapply(fit, coef)`

`mu.link`and sigma may be employed depending on covariates through the argument

`sigma.formula`. The

`vasicekreg`package is available online at https://cloud.r-project.org/web/packages/vasicekreg/index.html (accessed on 24 March 2022) and through it we can also consider, beyond quantile regression, mean regression and all the functionality of the

`gamlss`package. For the other fitted models, the corresponding ML estimates were calculated with the

`unitquantreg`package, which is under development and available online at https://github.com/AndrMenezes/unitquantreg (accessed on 24 March 2022).

#### 5.2. The VASI Mean Regression Model

`baquantreg`package of

`R`[45]. The response corresponds to the proportion of votes in the 2014 presidential election in Brazil, obtained by the elected president, Dilma Rousseff, in the Minas Gerais and Piaui states. First, for each state, we considered the Human Development Index (HDI) in 2010, centered at the mean, as covariate and a regression structure given by

`R`codes as follows:

`vasi <- gamlss(percvotes ~ state + hdi, sigma.formula = ~ state,`

`data = votesmgpi, family = VASIM(), trace = FALSE)`

`beta <- gamlss(percvotes ~ state + hdi, sigma.formula = ~ state,`

`data = votesmgpi, family = BE(), trace = FALSE)`

`simp <- gamlss(percvotes ~ state + hdi, sigma.formula = ~ state,`

`data = votesmgpi, family = SIMPLEX(), trace = FALSE)`

`GAIC(vasi, beta, simp)`

`DF AIC`

`vasi 5 -1887.173`

`beta 5 -1882.968`

`simp 5 -1877.755`

`plot`function, applied to objects

`vasi`,

`beta`, or

`simp`, we have available four plots for checking the normalized quantile residuals. The four plots are residuals against the fitted values of the parameter $\alpha $, residuals against an index, a kernel density estimate of the residuals, and the QQ plot of the residuals.

## 6. Concluding Remarks

`R`software, which was implemented by the authors of this paper.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Other Distributions

- The ULOG distribution [46] is obtained from the transformation$$Y=\frac{exp\left(\frac{X-\alpha}{\theta}\right)}{1+exp\left(\frac{X-\alpha}{\theta}\right)},$$$$f(y;\alpha ,\theta )=\frac{\theta exp\left(\alpha \right){\left(\right)}^{\frac{y}{1-y}}\theta -1}{}{\left(\right)}^{1}$$$$F(y;\alpha ,\theta )=\frac{exp\left(\alpha \right){\left(\right)}^{\frac{y}{1-y}}\theta}{}1+exp\left(\alpha \right){\left(\right)}^{\frac{y}{1-y}}\theta $$$$Q(\tau ;\alpha ,\theta )=\frac{exp\left(\right)open="("\; close=")">-\frac{\alpha}{\theta}}{{\left(\right)}^{\frac{\tau}{1-\tau}}}$$$$\alpha ={h}^{-1}\left(\mu \right)=log\left(\right)open="("\; close=")">\frac{\tau}{1-\tau}.$$
- The JOSB distribution [48] can be obtained from the transformation$$Y=\frac{exp\left(\frac{X-\alpha}{\theta}\right)}{1+exp\left(\frac{X-\alpha}{\theta}\right)},$$$$f(y;\alpha ,\theta )={\displaystyle \frac{\theta}{\sqrt{2\pi}}}\frac{1}{y(1-y)}exp\left(\right)open="\{"\; close="\}">-{\displaystyle \frac{1}{2}}{\left(\right)}^{\alpha}$$$$F(y;\alpha ,\theta )=\mathrm{\Phi}\left(\right)open="["\; close="]">\alpha +\theta log\left({\displaystyle \frac{y}{1-y}}\right)$$$$Q(\tau ;\alpha ,\theta )={\displaystyle \frac{exp\left[{\displaystyle \frac{{\mathrm{\Phi}}^{-1}\left(\tau \right)-\alpha}{\theta}}\right]}{1+exp\left[{\displaystyle \frac{{\mathrm{\Phi}}^{-1}\left(\tau \right)-\alpha}{\theta}}\right]}},$$$$\alpha ={h}^{-1}\left(\mu \right)={\mathrm{\Phi}}^{-1}\left(\tau \right)-\theta log\left({\displaystyle \frac{\mu}{1-\mu}}\right).$$
- The KUMA distribution [49] can be obtained from the transformation $Y=exp(-X)$, where $X\sim \mathrm{EE}(\alpha ,\theta )$ denotes a random variable with exponentiated–exponential distribution. The associated PDF, CDF, and QF are written, respectively, as$$f(y;\alpha ,\theta )=\alpha \theta {y}^{\theta -1}{(1-{y}^{\theta})}^{\alpha -1},$$$$F(y;\alpha ,\theta )=1-{\left(\right)}^{1}\alpha $$$$Q(\tau ;\alpha ,\theta )={\left(\right)}^{1},$$$$\alpha ={h}^{-1}\left(\mu \right)=\frac{log(1-\tau )}{log(1-{\mu}^{\theta})}.$$
- The UWEI distribution [18,19] is obtained from the transformation $Y=exp(-X)$, where $X\sim \mathrm{WEI}(\alpha ,\theta )$ denotes a random variable with Weibull distribution [50]. The associated PDF, CDF, and QF are written, respectively, as$$f(y;\alpha ,\theta )=\frac{\alpha \theta}{y}{\left(\right)}^{-}\theta -1$$$$F(y;\alpha ,\theta )=exp\left(\right)open="\{"\; close="\}">-\alpha {\left(\right)}^{-}\theta $$$$Q\left(\right)open="("\; close=")">\tau ;\alpha ,\theta $$$$\alpha ={h}^{-1}\left(\mu \right)=-\frac{log\left(\tau \right)}{{[-log\left(\mu \right)]}^{\theta}}.$$
- The beta distribution parametrized in terms of its mean and dispersion parameters was given in [43]. The corresponding PDF is written as$$f(y;\alpha ,\theta )=\frac{\mathrm{\Gamma}\left(\theta \right)}{\mathrm{\Gamma}\left(\theta \alpha \right)\mathrm{\Gamma}\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">1-\alpha}{y}^{\alpha \theta -1}{(1-y)}^{\left(\right)}$$

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**Figure 2.**Plots of the PDF stated in (1) with quantile parameterization for different combinations of $\theta $, $\mu $, and $\tau $.

**Figure 3.**Estimated quantile process plot for ${\delta}_{i}$, with $i\in \{0,1,2,3,4,5\}$ and $\theta $.

**Figure 4.**QQ plots with simulated envelopes of Cox–Snell (first row) and normalized quantile (second row) residuals for the listed $\tau $ quantile with fat proportion in the arms.

**Table 1.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=2.00$, and $\theta =0.25$) with simulated data.

Bias | RMSE | CP_{95%} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | θ | δ0 | δ1 | θ | δ0 | δ1 | θ |

0.10 | 20 | $0.0897$ | $0.0210$ | $-0.0227$ | $0.3373$ | $0.2856$ | $0.0623$ | $89.88$ | $91.88$ | $85.16$ |

50 | $0.0361$ | $0.0087$ | $-0.0085$ | $0.2068$ | $0.1852$ | $0.0382$ | $93.54$ | $93.56$ | $91.40$ | |

80 | $0.0218$ | $0.0047$ | $-0.0050$ | $0.1653$ | $0.1361$ | $0.0299$ | $93.94$ | $93.72$ | $92.84$ | |

110 | $0.0153$ | $0.0033$ | $-0.0036$ | $0.1406$ | $0.1154$ | $0.0252$ | $94.10$ | $93.90$ | $93.62$ | |

140 | $0.0131$ | $0.0024$ | $-0.0029$ | $0.1244$ | $0.1014$ | $0.0222$ | $94.26$ | $94.38$ | $94.02$ | |

170 | $0.0101$ | $0.0018$ | $-0.0022$ | $0.1117$ | $0.0921$ | $0.0202$ | $94.66$ | $94.80$ | $94.38$ | |

200 | $0.0090$ | $0.0025$ | $-0.0019$ | $0.1027$ | $0.0854$ | $0.0184$ | $94.64$ | $94.66$ | $94.66$ | |

0.25 | 20 | $0.0483$ | $0.0132$ | $-0.0228$ | $0.2697$ | $0.2829$ | $0.0624$ | $91.34$ | $91.78$ | $85.14$ |

50 | $0.0209$ | $0.0063$ | $-0.0086$ | $0.1682$ | $0.1845$ | $0.0384$ | $94.18$ | $93.54$ | $91.32$ | |

80 | $0.0129$ | $0.0035$ | $-0.0051$ | $0.1353$ | $0.1357$ | $0.0301$ | $94.20$ | $93.74$ | $92.80$ | |

110 | $0.0088$ | $0.0025$ | $-0.0037$ | $0.1154$ | $0.1150$ | $0.0254$ | $94.38$ | $93.98$ | $93.44$ | |

140 | $0.0079$ | $0.0017$ | $-0.0030$ | $0.1026$ | $0.1011$ | $0.0224$ | $94.74$ | $94.28$ | $94.02$ | |

170 | $0.0061$ | $0.0013$ | $-0.0023$ | $0.0918$ | $0.0918$ | $0.0204$ | $94.66$ | $94.84$ | $94.38$ | |

200 | $0.0056$ | $0.0021$ | $-0.0020$ | $0.0848$ | $0.0852$ | $0.0186$ | $94.66$ | $94.64$ | $94.58$ | |

0.50 | 20 | $0.0030$ | $0.0059$ | $-0.0228$ | $0.2371$ | $0.2813$ | $0.0625$ | $93.00$ | $91.68$ | $85.08$ |

50 | $0.0042$ | $0.0042$ | $-0.0086$ | $0.1498$ | $0.1841$ | $0.0385$ | $94.72$ | $93.62$ | $91.42$ | |

80 | $0.0029$ | $0.0024$ | $-0.0051$ | $0.1208$ | $0.1354$ | $0.0302$ | $94.56$ | $93.84$ | $92.68$ | |

110 | $0.0016$ | $0.0017$ | $-0.0037$ | $0.1032$ | $0.1148$ | $0.0255$ | $94.74$ | $94.06$ | $93.52$ | |

140 | $0.0021$ | $0.0011$ | $-0.0030$ | $0.0922$ | $0.1008$ | $0.0224$ | $95.18$ | $94.36$ | $93.92$ | |

170 | $0.0017$ | $0.0009$ | $-0.0023$ | $0.0824$ | $0.0916$ | $0.0205$ | $94.94$ | $94.78$ | $94.10$ | |

200 | $0.0017$ | $0.0017$ | $-0.0020$ | $0.0765$ | $0.0851$ | $0.0186$ | $94.94$ | $94.72$ | $94.68$ | |

0.75 | 20 | $-0.0410$ | $0.0002$ | $-0.0226$ | $0.2643$ | $0.2811$ | $0.0624$ | $91.52$ | $91.68$ | $85.28$ |

50 | $-0.0120$ | $0.0026$ | $-0.0085$ | $0.1660$ | $0.1841$ | $0.0385$ | $94.38$ | $93.48$ | $91.20$ | |

80 | $-0.0069$ | $0.0017$ | $-0.0051$ | $0.1332$ | $0.1354$ | $0.0302$ | $94.78$ | $93.84$ | $92.54$ | |

110 | $-0.0055$ | $0.0012$ | $-0.0037$ | $0.1135$ | $0.1147$ | $0.0255$ | $94.56$ | $94.24$ | $93.44$ | |

140 | $-0.0036$ | $0.0007$ | $-0.0030$ | $0.1013$ | $0.1008$ | $0.0224$ | $95.04$ | $94.38$ | $93.90$ | |

170 | $-0.0028$ | $0.0007$ | $-0.0023$ | $0.0909$ | $0.0915$ | $0.0204$ | $95.12$ | $94.72$ | $94.12$ | |

200 | $-0.0021$ | $0.0015$ | $-0.0020$ | $0.0841$ | $0.0850$ | $0.0186$ | $94.78$ | $94.78$ | $94.60$ | |

0.90 | 20 | $-0.0790$ | $-0.0037$ | $-0.0224$ | $0.3272$ | $0.2818$ | $0.0621$ | $89.80$ | $91.82$ | $85.34$ |

50 | $-0.0260$ | $0.0016$ | $-0.0085$ | $0.2033$ | $0.1846$ | $0.0383$ | $93.78$ | $93.26$ | $91.24$ | |

80 | $-0.0154$ | $0.0014$ | $-0.0051$ | $0.1622$ | $0.1355$ | $0.0300$ | $94.36$ | $93.86$ | $92.56$ | |

110 | $-0.0116$ | $0.0010$ | $-0.0036$ | $0.1380$ | $0.1148$ | $0.0254$ | $94.20$ | $94.32$ | $93.20$ | |

140 | $-0.0085$ | $0.0005$ | $-0.0030$ | $0.1226$ | $0.1009$ | $0.0223$ | $94.20$ | $94.40$ | $94.06$ | |

170 | $-0.0066$ | $0.0005$ | $-0.0023$ | $0.1105$ | $0.0915$ | $0.0204$ | $94.80$ | $94.72$ | $94.08$ | |

200 | $-0.0055$ | $0.0013$ | $-0.0020$ | $0.1017$ | $0.0851$ | $0.0186$ | $94.82$ | $95.00$ | $94.64$ |

**Table 2.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=2.00$, and $\theta =0.50$) with simulated data.

Bias | RMSE | CP_{95%} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | θ | δ0 | δ1 | θ | δ0 | δ1 | θ |

0.10 | 20 | $0.1632$ | $0.0498$ | $-0.0381$ | $0.5939$ | $0.5017$ | $0.0911$ | $90.32$ | $91.80$ | $87.54$ |

50 | $0.0658$ | $0.0207$ | $-0.0143$ | $0.3606$ | $0.3223$ | $0.0529$ | $93.88$ | $93.60$ | $92.34$ | |

80 | $0.0396$ | $0.0114$ | $-0.0084$ | $0.2872$ | $0.2362$ | $0.0408$ | $94.14$ | $93.76$ | $93.66$ | |

110 | $0.0280$ | $0.0082$ | $-0.0061$ | $0.2441$ | $0.2003$ | $0.0342$ | $94.29$ | $94.01$ | $93.97$ | |

140 | $0.0239$ | $0.0059$ | $-0.0049$ | $0.2160$ | $0.1759$ | $0.0301$ | $94.35$ | $94.29$ | $94.41$ | |

170 | $0.0183$ | $0.0046$ | $-0.0038$ | $0.1938$ | $0.1597$ | $0.0273$ | $94.77$ | $94.79$ | $94.67$ | |

200 | $0.0163$ | $0.0055$ | $-0.0033$ | $0.1784$ | $0.1482$ | $0.0248$ | $94.70$ | $94.64$ | $94.87$ | |

0.25 | 20 | $0.0876$ | $0.0350$ | $-0.0379$ | $0.4727$ | $0.4952$ | $0.0914$ | $91.84$ | $91.74$ | $87.56$ |

50 | $0.0382$ | $0.0163$ | $-0.0142$ | $0.2926$ | $0.3208$ | $0.0532$ | $94.22$ | $93.36$ | $92.48$ | |

80 | $0.0234$ | $0.0092$ | $-0.0084$ | $0.2349$ | $0.2355$ | $0.0410$ | $94.38$ | $93.58$ | $93.34$ | |

110 | $0.0161$ | $0.0066$ | $-0.0061$ | $0.2002$ | $0.1995$ | $0.0345$ | $94.40$ | $94.06$ | $93.84$ | |

140 | $0.0144$ | $0.0048$ | $-0.0049$ | $0.1780$ | $0.1753$ | $0.0303$ | $94.76$ | $94.28$ | $94.40$ | |

170 | $0.0111$ | $0.0038$ | $-0.0038$ | $0.1592$ | $0.1592$ | $0.0275$ | $94.70$ | $94.84$ | $94.70$ | |

200 | $0.0101$ | $0.0048$ | $-0.0032$ | $0.1472$ | $0.1477$ | $0.0250$ | $94.62$ | $94.74$ | $94.70$ | |

0.50 | 20 | $0.0080$ | $0.0214$ | $-0.0379$ | $0.4137$ | $0.4911$ | $0.0915$ | $93.06$ | $91.64$ | $87.60$ |

50 | $0.0090$ | $0.0123$ | $-0.0142$ | $0.2603$ | $0.3199$ | $0.0533$ | $94.92$ | $93.30$ | $92.64$ | |

80 | $0.0059$ | $0.0072$ | $-0.0085$ | $0.2096$ | $0.2350$ | $0.0412$ | $94.62$ | $93.88$ | $93.44$ | |

110 | $0.0035$ | $0.0052$ | $-0.0061$ | $0.1789$ | $0.1991$ | $0.0346$ | $94.78$ | $94.24$ | $93.74$ | |

140 | $0.0042$ | $0.0037$ | $-0.0049$ | $0.1599$ | $0.1748$ | $0.0303$ | $95.26$ | $94.36$ | $94.44$ | |

170 | $0.0033$ | $0.0030$ | $-0.0038$ | $0.1429$ | $0.1588$ | $0.0276$ | $95.00$ | $94.78$ | $94.68$ | |

200 | $0.0034$ | $0.0041$ | $-0.0033$ | $0.1325$ | $0.1474$ | $0.0251$ | $94.96$ | $94.84$ | $94.84$ | |

0.75 | 20 | $-0.0679$ | $0.0124$ | $-0.0378$ | $0.4600$ | $0.4900$ | $0.0914$ | $91.62$ | $91.66$ | $87.72$ |

50 | $-0.0189$ | $0.0098$ | $-0.0141$ | $0.2882$ | $0.3199$ | $0.0533$ | $94.44$ | $93.28$ | $92.58$ | |

80 | $-0.0109$ | $0.0062$ | $-0.0084$ | $0.2309$ | $0.2348$ | $0.0412$ | $94.84$ | $93.98$ | $93.44$ | |

110 | $-0.0087$ | $0.0044$ | $-0.0061$ | $0.1968$ | $0.1990$ | $0.0346$ | $94.60$ | $94.48$ | $93.74$ | |

140 | $-0.0056$ | $0.0031$ | $-0.0049$ | $0.1757$ | $0.1747$ | $0.0303$ | $95.08$ | $94.30$ | $94.34$ | |

170 | $-0.0043$ | $0.0026$ | $-0.0038$ | $0.1576$ | $0.1587$ | $0.0275$ | $95.10$ | $94.66$ | $94.60$ | |

200 | $-0.0032$ | $0.0038$ | $-0.0033$ | $0.1457$ | $0.1473$ | $0.0251$ | $94.78$ | $94.84$ | $94.88$ | |

0.90 | 20 | $-0.1326$ | $0.0082$ | $-0.0374$ | $0.5692$ | $0.4912$ | $0.0910$ | $89.94$ | $91.68$ | $87.84$ |

50 | $-0.0426$ | $0.0090$ | $-0.0140$ | $0.3530$ | $0.3207$ | $0.0530$ | $93.96$ | $93.10$ | $92.58$ | |

80 | $-0.0253$ | $0.0062$ | $-0.0084$ | $0.2813$ | $0.2351$ | $0.0410$ | $94.38$ | $93.98$ | $93.36$ | |

110 | $-0.0190$ | $0.0044$ | $-0.0060$ | $0.2392$ | $0.1992$ | $0.0344$ | $94.24$ | $94.32$ | $93.88$ | |

140 | $-0.0140$ | $0.0030$ | $-0.0048$ | $0.2127$ | $0.1749$ | $0.0302$ | $94.32$ | $94.44$ | $94.26$ | |

170 | $-0.0108$ | $0.0027$ | $-0.0038$ | $0.1916$ | $0.1587$ | $0.0274$ | $94.86$ | $94.70$ | $94.52$ | |

200 | $-0.0089$ | $0.0038$ | $-0.0033$ | $0.1763$ | $0.1474$ | $0.0250$ | $94.78$ | $95.02$ | $94.80$ |

**Table 3.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=2.00$, and $\theta =0.75$) with simulated data.

Bias | RMSE | CP_{95%} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | θ | δ0 | δ1 | θ | δ0 | δ1 | θ |

0.10 | 20 | $0.3389$ | $-0.0624$ | $-0.0548$ | $1.0520$ | $0.8201$ | $0.0901$ | $90.03$ | $91.58$ | $89.31$ |

50 | $0.1364$ | $-0.1039$ | $-0.0287$ | $0.6283$ | $0.5316$ | $0.0495$ | $93.43$ | $92.54$ | $91.61$ | |

80 | $0.0863$ | $-0.1028$ | $-0.0226$ | $0.5013$ | $0.4020$ | $0.0378$ | $92.86$ | $91.49$ | $91.62$ | |

110 | $0.0792$ | $-0.1130$ | $-0.0214$ | $0.4327$ | $0.3528$ | $0.0336$ | $93.45$ | $90.92$ | $91.34$ | |

140 | $0.0775$ | $-0.1020$ | $-0.0198$ | $0.3766$ | $0.3145$ | $0.0306$ | $92.86$ | $89.44$ | $89.60$ | |

170 | $0.0642$ | $-0.1081$ | $-0.0179$ | $0.3309$ | $0.2995$ | $0.0274$ | $94.08$ | $89.10$ | $89.57$ | |

200 | $0.0685$ | $-0.1129$ | $-0.0166$ | $0.3096$ | $0.2849$ | $0.0249$ | $93.95$ | $87.90$ | $90.39$ | |

0.25 | 20 | $0.1600$ | $0.0259$ | $-0.0418$ | $0.8327$ | $0.8425$ | $0.0816$ | $92.49$ | $91.78$ | $91.34$ |

50 | $0.0615$ | $-0.0166$ | $-0.0179$ | $0.5087$ | $0.5411$ | $0.0438$ | $94.29$ | $93.27$ | $93.71$ | |

80 | $0.0396$ | $-0.0205$ | $-0.0124$ | $0.4073$ | $0.4021$ | $0.0331$ | $94.39$ | $93.17$ | $94.07$ | |

110 | $0.0287$ | $-0.0266$ | $-0.0104$ | $0.3446$ | $0.3363$ | $0.0279$ | $94.50$ | $93.82$ | $94.07$ | |

140 | $0.0244$ | $-0.0301$ | $-0.0092$ | $0.3050$ | $0.2967$ | $0.0243$ | $95.25$ | $94.09$ | $94.09$ | |

170 | $0.0144$ | $-0.0312$ | $-0.0083$ | $0.2717$ | $0.2721$ | $0.0220$ | $95.06$ | $94.38$ | $94.69$ | |

200 | $0.0089$ | $-0.0284$ | $-0.0077$ | $0.2517$ | $0.2524$ | $0.0201$ | $94.49$ | $94.45$ | $94.66$ | |

0.50 | 20 | $0.0165$ | $0.0530$ | $-0.0372$ | $0.7268$ | $0.8619$ | $0.0799$ | $93.60$ | $91.61$ | $91.44$ |

50 | $0.0124$ | $0.0229$ | $-0.0141$ | $0.4513$ | $0.5546$ | $0.0426$ | $95.06$ | $93.03$ | $94.06$ | |

80 | $0.0065$ | $0.0104$ | $-0.0087$ | $0.3622$ | $0.4062$ | $0.0323$ | $94.81$ | $93.89$ | $94.10$ | |

110 | $0.0018$ | $0.0043$ | $-0.0065$ | $0.3088$ | $0.3422$ | $0.0268$ | $94.90$ | $94.47$ | $94.50$ | |

140 | $0.0028$ | $0.0019$ | $-0.0053$ | $0.2755$ | $0.3020$ | $0.0233$ | $95.50$ | $94.36$ | $94.66$ | |

170 | $0.0020$ | $0.0016$ | $-0.0044$ | $0.2475$ | $0.2749$ | $0.0212$ | $95.04$ | $94.56$ | $95.00$ | |

200 | $0.0026$ | $0.0024$ | $-0.0038$ | $0.2296$ | $0.2550$ | $0.0192$ | $95.02$ | $94.80$ | $95.28$ | |

0.75 | 20 | $-0.1108$ | $0.0548$ | $-0.0358$ | $0.8099$ | $0.8653$ | $0.0794$ | $92.13$ | $91.49$ | $91.33$ |

50 | $-0.0285$ | $0.0315$ | $-0.0130$ | $0.5028$ | $0.5590$ | $0.0424$ | $94.56$ | $93.06$ | $93.96$ | |

80 | $-0.0169$ | $0.0191$ | $-0.0077$ | $0.4015$ | $0.4083$ | $0.0320$ | $94.80$ | $93.89$ | $93.97$ | |

110 | $-0.0141$ | $0.0138$ | $-0.0056$ | $0.3418$ | $0.3460$ | $0.0266$ | $94.71$ | $94.41$ | $94.49$ | |

140 | $-0.0093$ | $0.0100$ | $-0.0045$ | $0.3048$ | $0.3037$ | $0.0232$ | $95.15$ | $94.34$ | $94.60$ | |

170 | $-0.0071$ | $0.0081$ | $-0.0035$ | $0.2734$ | $0.2756$ | $0.0210$ | $95.12$ | $94.59$ | $95.04$ | |

200 | $-0.0052$ | $0.0095$ | $-0.0030$ | $0.2527$ | $0.2561$ | $0.0191$ | $94.79$ | $94.83$ | $95.13$ | |

0.90 | 20 | $-0.2196$ | $0.0570$ | $-0.0353$ | $1.0056$ | $0.8707$ | $0.0791$ | $90.56$ | $91.60$ | $91.38$ |

50 | $-0.0670$ | $0.0343$ | $-0.0127$ | $0.6171$ | $0.5615$ | $0.0423$ | $94.00$ | $93.00$ | $93.94$ | |

80 | $-0.0397$ | $0.0218$ | $-0.0076$ | $0.4900$ | $0.4101$ | $0.0319$ | $94.54$ | $93.90$ | $94.02$ | |

110 | $-0.0301$ | $0.0157$ | $-0.0054$ | $0.4161$ | $0.3468$ | $0.0265$ | $94.28$ | $94.42$ | $94.22$ | |

140 | $-0.0219$ | $0.0116$ | $-0.0043$ | $0.3697$ | $0.3041$ | $0.0232$ | $94.50$ | $94.34$ | $94.54$ | |

170 | $-0.0170$ | $0.0098$ | $-0.0034$ | $0.3327$ | $0.2757$ | $0.0209$ | $94.88$ | $94.62$ | $95.00$ | |

200 | $-0.0139$ | $0.0111$ | $-0.0029$ | $0.3062$ | $0.2560$ | $0.0190$ | $94.96$ | $95.10$ | $95.14$ |

**Table 4.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=1.00$, ${\delta}_{2}=0.50$, and $\theta =0.25$) with simulated data.

Bias | RMSE | CP_{95%} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ |

0.10 | 20 | $0.1236$ | $0.0166$ | $0.0169$ | $-0.0328$ | $0.4493$ | $0.4989$ | $0.3022$ | $0.0667$ | $89.44$ | $91.02$ | $91.14$ | $81.10$ |

50 | $0.0480$ | $0.0075$ | $0.0009$ | $-0.0122$ | $0.2612$ | $0.3027$ | $0.1438$ | $0.0396$ | $92.98$ | $93.78$ | $93.54$ | $89.34$ | |

80 | $0.0299$ | $0.0035$ | $0.0017$ | $-0.0074$ | $0.2114$ | $0.2402$ | $0.1119$ | $0.0308$ | $93.44$ | $93.92$ | $93.94$ | $91.56$ | |

110 | $0.0199$ | $0.0047$ | $0.0012$ | $-0.0054$ | $0.1745$ | $0.2038$ | $0.0958$ | $0.0258$ | $94.20$ | $94.42$ | $94.26$ | $92.94$ | |

140 | $0.0157$ | $0.0058$ | $0.0014$ | $-0.0043$ | $0.1501$ | $0.1781$ | $0.0820$ | $0.0227$ | $94.40$ | $94.40$ | $94.38$ | $93.30$ | |

170 | $0.0121$ | $0.0055$ | $0.0009$ | $-0.0034$ | $0.1308$ | $0.1615$ | $0.0750$ | $0.0206$ | $94.60$ | $94.64$ | $94.82$ | $93.66$ | |

200 | $0.0110$ | $0.0043$ | $0.0013$ | $-0.0030$ | $0.1209$ | $0.1486$ | $0.0703$ | $0.0187$ | $94.54$ | $94.54$ | $94.92$ | $94.42$ | |

0.25 | 20 | $0.0680$ | $0.0086$ | $0.0129$ | $-0.0329$ | $0.3985$ | $0.4942$ | $0.2995$ | $0.0667$ | $90.90$ | $91.08$ | $91.42$ | $81.14$ |

50 | $0.0279$ | $0.0044$ | $-0.0006$ | $-0.0122$ | $0.2336$ | $0.3014$ | $0.1432$ | $0.0396$ | $93.46$ | $93.82$ | $93.58$ | $89.34$ | |

80 | $0.0177$ | $0.0017$ | $0.0008$ | $-0.0074$ | $0.1907$ | $0.2396$ | $0.1115$ | $0.0308$ | $93.92$ | $93.92$ | $93.88$ | $91.54$ | |

110 | $0.0111$ | $0.0034$ | $0.0006$ | $-0.0054$ | $0.1576$ | $0.2032$ | $0.0956$ | $0.0258$ | $94.32$ | $94.34$ | $94.30$ | $92.96$ | |

140 | $0.0086$ | $0.0047$ | $0.0008$ | $-0.0043$ | $0.1353$ | $0.1777$ | $0.0818$ | $0.0227$ | $94.78$ | $94.52$ | $94.44$ | $93.28$ | |

170 | $0.0065$ | $0.0047$ | $0.0005$ | $-0.0034$ | $0.1168$ | $0.1611$ | $0.0749$ | $0.0206$ | $94.90$ | $94.58$ | $94.74$ | $93.64$ | |

200 | $0.0062$ | $0.0036$ | $0.0009$ | $-0.0030$ | $0.1080$ | $0.1483$ | $0.0702$ | $0.0187$ | $94.74$ | $94.52$ | $94.92$ | $94.48$ | |

0.50 | 20 | $0.0077$ | $0.0008$ | $0.0088$ | $-0.0329$ | $0.3749$ | $0.4900$ | $0.2970$ | $0.0667$ | $91.48$ | $91.06$ | $91.42$ | $81.22$ |

50 | $0.0060$ | $0.0014$ | $-0.0021$ | $-0.0122$ | $0.2213$ | $0.3004$ | $0.1428$ | $0.0396$ | $93.84$ | $93.80$ | $93.68$ | $89.38$ | |

80 | $0.0045$ | $-0.0001$ | $-0.0001$ | $-0.0074$ | $0.1812$ | $0.2391$ | $0.1113$ | $0.0308$ | $93.64$ | $93.90$ | $93.84$ | $91.44$ | |

110 | $0.0016$ | $0.0021$ | $-0.0001$ | $-0.0054$ | $0.1505$ | $0.2028$ | $0.0955$ | $0.0258$ | $94.20$ | $94.34$ | $94.16$ | $93.02$ | |

140 | $0.0009$ | $0.0037$ | $0.0003$ | $-0.0043$ | $0.1290$ | $0.1774$ | $0.0817$ | $0.0227$ | $94.92$ | $94.58$ | $94.46$ | $93.34$ | |

170 | $0.0005$ | $0.0039$ | $0.0002$ | $-0.0034$ | $0.1109$ | $0.1609$ | $0.0749$ | $0.0206$ | $94.94$ | $94.62$ | $94.84$ | $93.68$ | |

200 | $0.0009$ | $0.0029$ | $0.0006$ | $-0.0030$ | $0.1024$ | $0.1482$ | $0.0701$ | $0.0187$ | $94.94$ | $94.56$ | $94.90$ | $94.52$ | |

0.75 | 20 | $-0.0513$ | $-0.0059$ | $0.0053$ | $-0.0329$ | $0.3901$ | $0.4870$ | $0.2950$ | $0.0667$ | $90.66$ | $91.12$ | $91.34$ | $81.14$ |

50 | $-0.0154$ | $-0.0012$ | $-0.0033$ | $-0.0122$ | $0.2308$ | $0.2998$ | $0.1425$ | $0.0396$ | $93.50$ | $93.90$ | $93.54$ | $89.28$ | |

80 | $-0.0085$ | $-0.0016$ | $-0.0008$ | $-0.0074$ | $0.1877$ | $0.2389$ | $0.1112$ | $0.0308$ | $94.12$ | $93.88$ | $93.92$ | $91.36$ | |

110 | $-0.0078$ | $0.0010$ | $-0.0006$ | $-0.0054$ | $0.1568$ | $0.2026$ | $0.0955$ | $0.0258$ | $94.30$ | $94.50$ | $94.14$ | $92.94$ | |

140 | $-0.0066$ | $0.0028$ | $-0.0001$ | $-0.0043$ | $0.1350$ | $0.1772$ | $0.0817$ | $0.0227$ | $94.70$ | $94.60$ | $94.50$ | $93.34$ | |

170 | $-0.0055$ | $0.0032$ | $-0.0002$ | $-0.0034$ | $0.1166$ | $0.1608$ | $0.0749$ | $0.0206$ | $94.70$ | $94.54$ | $94.80$ | $93.62$ | |

200 | $-0.0043$ | $0.0023$ | $0.0003$ | $-0.0030$ | $0.1072$ | $0.1482$ | $0.0702$ | $0.0187$ | $94.96$ | $94.50$ | $94.98$ | $94.58$ | |

0.90 | 20 | $-0.1034$ | $-0.0110$ | $0.0025$ | $-0.0329$ | $0.4329$ | $0.4850$ | $0.2938$ | $0.0668$ | $89.26$ | $91.16$ | $91.18$ | $81.18$ |

50 | $-0.0342$ | $-0.0032$ | $-0.0043$ | $-0.0122$ | $0.2559$ | $0.2996$ | $0.1425$ | $0.0396$ | $92.90$ | $93.88$ | $93.48$ | $89.28$ | |

80 | $-0.0200$ | $-0.0028$ | $-0.0014$ | $-0.0074$ | $0.2060$ | $0.2390$ | $0.1113$ | $0.0308$ | $93.50$ | $93.84$ | $93.98$ | $91.38$ | |

110 | $-0.0161$ | $0.0001$ | $-0.0011$ | $-0.0054$ | $0.1729$ | $0.2026$ | $0.0955$ | $0.0258$ | $93.94$ | $94.56$ | $94.14$ | $92.92$ | |

140 | $-0.0132$ | $0.0021$ | $-0.0005$ | $-0.0043$ | $0.1496$ | $0.1772$ | $0.0818$ | $0.0227$ | $94.38$ | $94.62$ | $94.48$ | $93.36$ | |

170 | $-0.0107$ | $0.0026$ | $-0.0005$ | $-0.0034$ | $0.1304$ | $0.1609$ | $0.0750$ | $0.0206$ | $94.26$ | $94.52$ | $94.66$ | $93.64$ | |

200 | $-0.0088$ | $0.0019$ | $0.0001$ | $-0.0029$ | $0.1194$ | $0.1483$ | $0.0702$ | $0.0187$ | $94.74$ | $94.38$ | $95.04$ | $94.62$ |

**Table 5.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=1.00$, ${\delta}_{2}=0.50$, and $\theta =0.50$) with simulated data.

Bias | RMSE | CP_{95%} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ |

0.10 | 20 | $0.2321$ | $0.0301$ | $0.0366$ | $-0.0528$ | $0.8048$ | $0.8884$ | $0.5378$ | $0.1004$ | $90.32$ | $91.40$ | $91.32$ | $84.53$ |

50 | $0.0887$ | $0.0149$ | $0.0030$ | $-0.0192$ | $0.4578$ | $0.5297$ | $0.2520$ | $0.0555$ | $93.32$ | $93.86$ | $93.78$ | $90.98$ | |

80 | $0.0552$ | $0.0071$ | $0.0040$ | $-0.0116$ | $0.3693$ | $0.4189$ | $0.1952$ | $0.0423$ | $93.70$ | $94.08$ | $94.00$ | $92.46$ | |

110 | $0.0368$ | $0.0092$ | $0.0030$ | $-0.0084$ | $0.3038$ | $0.3546$ | $0.1669$ | $0.0352$ | $94.38$ | $94.44$ | $94.40$ | $93.40$ | |

140 | $0.0289$ | $0.0110$ | $0.0030$ | $-0.0067$ | $0.2611$ | $0.3097$ | $0.1426$ | $0.0308$ | $94.54$ | $94.52$ | $94.44$ | $93.82$ | |

170 | $0.0224$ | $0.0104$ | $0.0021$ | $-0.0053$ | $0.2273$ | $0.2806$ | $0.1303$ | $0.0278$ | $94.76$ | $94.64$ | $94.86$ | $94.34$ | |

200 | $0.0202$ | $0.0081$ | $0.0026$ | $-0.0046$ | $0.2100$ | $0.2581$ | $0.1220$ | $0.0253$ | $94.70$ | $94.52$ | $94.96$ | $94.74$ | |

0.25 | 20 | $0.1309$ | $0.0157$ | $0.0291$ | $-0.0527$ | $0.7083$ | $0.8733$ | $0.5293$ | $0.1004$ | $91.64$ | $91.34$ | $91.34$ | $84.44$ |

50 | $0.0525$ | $0.0094$ | $0.0002$ | $-0.0192$ | $0.4084$ | $0.5258$ | $0.2503$ | $0.0555$ | $93.70$ | $93.98$ | $93.72$ | $91.00$ | |

80 | $0.0334$ | $0.0038$ | $0.0023$ | $-0.0116$ | $0.3324$ | $0.4170$ | $0.1943$ | $0.0423$ | $94.02$ | $94.08$ | $93.94$ | $92.52$ | |

110 | $0.0211$ | $0.0069$ | $0.0018$ | $-0.0084$ | $0.2742$ | $0.3532$ | $0.1663$ | $0.0352$ | $94.46$ | $94.36$ | $94.38$ | $93.36$ | |

140 | $0.0163$ | $0.0091$ | $0.0021$ | $-0.0067$ | $0.2350$ | $0.3086$ | $0.1422$ | $0.0308$ | $94.78$ | $94.58$ | $94.40$ | $93.80$ | |

170 | $0.0123$ | $0.0089$ | $0.0015$ | $-0.0053$ | $0.2028$ | $0.2797$ | $0.1301$ | $0.0278$ | $94.96$ | $94.68$ | $94.82$ | $94.30$ | |

200 | $0.0116$ | $0.0069$ | $0.0021$ | $-0.0046$ | $0.1875$ | $0.2575$ | $0.1218$ | $0.0253$ | $94.88$ | $94.56$ | $94.90$ | $94.80$ | |

0.50 | 20 | $0.0235$ | $0.0024$ | $0.0221$ | $-0.0528$ | $0.6615$ | $0.8605$ | $0.5219$ | $0.1004$ | $92.16$ | $91.34$ | $91.30$ | $84.38$ |

50 | $0.0141$ | $0.0041$ | $-0.0023$ | $-0.0192$ | $0.3858$ | $0.5227$ | $0.2489$ | $0.0555$ | $94.04$ | $93.98$ | $93.80$ | $90.96$ | |

80 | $0.0101$ | $0.0007$ | $0.0007$ | $-0.0116$ | $0.3152$ | $0.4156$ | $0.1937$ | $0.0423$ | $93.90$ | $94.00$ | $93.96$ | $92.52$ | |

110 | $0.0043$ | $0.0045$ | $0.0007$ | $-0.0084$ | $0.2614$ | $0.3521$ | $0.1659$ | $0.0352$ | $94.28$ | $94.38$ | $94.30$ | $93.36$ | |

140 | $0.0029$ | $0.0072$ | $0.0011$ | $-0.0067$ | $0.2240$ | $0.3078$ | $0.1419$ | $0.0308$ | $94.96$ | $94.60$ | $94.56$ | $93.86$ | |

170 | $0.0017$ | $0.0075$ | $0.0008$ | $-0.0053$ | $0.1924$ | $0.2792$ | $0.1300$ | $0.0278$ | $94.96$ | $94.66$ | $94.86$ | $94.30$ | |

200 | $0.0023$ | $0.0057$ | $0.0015$ | $-0.0046$ | $0.1776$ | $0.2571$ | $0.1217$ | $0.0253$ | $95.00$ | $94.56$ | $94.94$ | $94.82$ | |

0.75 | 20 | $-0.0798$ | $-0.0076$ | $0.0166$ | $-0.0528$ | $0.6849$ | $0.8515$ | $0.5167$ | $0.1005$ | $90.96$ | $91.40$ | $91.30$ | $84.42$ |

50 | $-0.0229$ | $-0.0001$ | $-0.0044$ | $-0.0192$ | $0.4016$ | $0.5209$ | $0.2481$ | $0.0555$ | $93.68$ | $93.96$ | $93.70$ | $91.06$ | |

80 | $-0.0124$ | $-0.0017$ | $-0.0005$ | $-0.0116$ | $0.3261$ | $0.4149$ | $0.1933$ | $0.0423$ | $94.18$ | $93.94$ | $94.10$ | $92.52$ | |

110 | $-0.0119$ | $0.0026$ | $-0.0002$ | $-0.0084$ | $0.2722$ | $0.3515$ | $0.1657$ | $0.0352$ | $94.36$ | $94.48$ | $94.20$ | $93.38$ | |

140 | $-0.0101$ | $0.0057$ | $0.0004$ | $-0.0067$ | $0.2341$ | $0.3074$ | $0.1418$ | $0.0308$ | $94.76$ | $94.68$ | $94.58$ | $93.86$ | |

170 | $-0.0085$ | $0.0063$ | $0.0002$ | $-0.0053$ | $0.2022$ | $0.2789$ | $0.1300$ | $0.0278$ | $94.70$ | $94.56$ | $94.80$ | $94.30$ | |

200 | $-0.0065$ | $0.0047$ | $0.0010$ | $-0.0046$ | $0.1858$ | $0.2570$ | $0.1217$ | $0.0252$ | $94.98$ | $94.54$ | $95.00$ | $94.76$ | |

0.90 | 20 | $-0.1701$ | $-0.0138$ | $0.0129$ | $-0.0528$ | $0.7586$ | $0.8466$ | $0.5137$ | $0.1005$ | $89.64$ | $91.48$ | $91.30$ | $84.52$ |

50 | $-0.0551$ | $-0.0029$ | $-0.0057$ | $-0.0192$ | $0.4449$ | $0.5202$ | $0.2479$ | $0.0555$ | $93.16$ | $94.02$ | $93.70$ | $91.08$ | |

80 | $-0.0319$ | $-0.0033$ | $-0.0013$ | $-0.0116$ | $0.3577$ | $0.4148$ | $0.1933$ | $0.0423$ | $93.58$ | $93.98$ | $94.12$ | $92.42$ | |

110 | $-0.0259$ | $0.0013$ | $-0.0009$ | $-0.0084$ | $0.3000$ | $0.3514$ | $0.1657$ | $0.0352$ | $94.02$ | $94.64$ | $94.28$ | $93.42$ | |

140 | $-0.0214$ | $0.0047$ | $-0.0001$ | $-0.0067$ | $0.2593$ | $0.3073$ | $0.1419$ | $0.0308$ | $94.48$ | $94.64$ | $94.58$ | $93.86$ | |

170 | $-0.0174$ | $0.0055$ | $-0.0002$ | $-0.0053$ | $0.2261$ | $0.2790$ | $0.1302$ | $0.0278$ | $94.26$ | $94.56$ | $94.66$ | $94.38$ | |

200 | $-0.0143$ | $0.0041$ | $0.0006$ | $-0.0046$ | $0.2069$ | $0.2571$ | $0.1219$ | $0.0252$ | $94.80$ | $94.40$ | $95.04$ | $94.80$ |

**Table 6.**Empirical bias, RMSE, and $95\%$ CP (true values: ${\delta}_{0}=1.00$, ${\delta}_{1}=1.00$, ${\delta}_{2}=0.50$, and $\theta =0.75$) with simulated data.

Bias | RMSE | CP_{95%} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

τ | n | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ | δ0 | δ1 | δ2 | θ |

0.10 | 20 | $0.5589$ | $-0.0299$ | $0.0406$ | $-0.0676$ | $1.5145$ | $1.5813$ | $0.9422$ | $0.1010$ | $91.22$ | $91.87$ | $91.65$ | $85.88$ |

50 | $0.2687$ | $-0.0555$ | $-0.0201$ | $-0.0347$ | $0.8187$ | $0.9035$ | $0.4341$ | $0.0537$ | $93.20$ | $94.52$ | $93.80$ | $89.33$ | |

80 | $0.1991$ | $-0.0373$ | $-0.0260$ | $-0.0276$ | $0.6558$ | $0.6952$ | $0.3317$ | $0.0416$ | $93.07$ | $94.50$ | $93.82$ | $89.55$ | |

110 | $0.1848$ | $-0.0319$ | $-0.0285$ | $-0.0253$ | $0.5474$ | $0.5877$ | $0.2916$ | $0.0362$ | $93.11$ | $94.93$ | $93.40$ | $87.85$ | |

140 | $0.1681$ | $-0.0421$ | $-0.0259$ | $-0.0231$ | $0.4713$ | $0.5131$ | $0.2539$ | $0.0326$ | $94.03$ | $95.76$ | $93.50$ | $87.67$ | |

170 | $0.1363$ | $-0.0290$ | $-0.0256$ | $-0.0216$ | $0.3935$ | $0.4508$ | $0.2241$ | $0.0301$ | $94.22$ | $97.11$ | $95.31$ | $86.28$ | |

200 | $0.1349$ | $-0.0166$ | $-0.0213$ | $-0.0215$ | $0.3682$ | $0.4068$ | $0.2050$ | $0.0284$ | $93.78$ | $96.41$ | $95.45$ | $85.89$ | |

0.25 | 20 | $0.2759$ | $0.0103$ | $0.0545$ | $-0.0537$ | $1.3020$ | $1.5745$ | $0.9434$ | $0.0921$ | $93.08$ | $92.02$ | $92.13$ | $88.39$ |

50 | $0.1102$ | $0.0043$ | $-0.0066$ | $-0.0216$ | $0.7261$ | $0.9258$ | $0.4359$ | $0.0462$ | $94.05$ | $94.07$ | $94.09$ | $92.99$ | |

80 | $0.0721$ | $-0.0058$ | $-0.0015$ | $-0.0147$ | $0.5838$ | $0.7325$ | $0.3361$ | $0.0344$ | $94.33$ | $93.83$ | $94.51$ | $93.68$ | |

110 | $0.0480$ | $0.0050$ | $-0.0054$ | $-0.0118$ | $0.4750$ | $0.6154$ | $0.2878$ | $0.0285$ | $94.85$ | $94.28$ | $94.47$ | $93.65$ | |

140 | $0.0409$ | $0.0030$ | $-0.0061$ | $-0.0100$ | $0.4093$ | $0.5379$ | $0.2415$ | $0.0248$ | $95.05$ | $94.41$ | $95.02$ | $94.18$ | |

170 | $0.0323$ | $0.0024$ | $-0.0075$ | $-0.0085$ | $0.3526$ | $0.4799$ | $0.2213$ | $0.0222$ | $94.99$ | $95.14$ | $95.21$ | $94.59$ | |

200 | $0.0298$ | $-0.0029$ | $-0.0053$ | $-0.0079$ | $0.3285$ | $0.4425$ | $0.2085$ | $0.0202$ | $94.48$ | $95.10$ | $95.30$ | $94.81$ | |

0.50 | 20 | $0.0663$ | $0.0057$ | $0.0547$ | $-0.0492$ | $1.2038$ | $1.5467$ | $0.9390$ | $0.0900$ | $93.42$ | $92.05$ | $92.37$ | $88.73$ |

50 | $0.0340$ | $0.0079$ | $-0.0049$ | $-0.0175$ | $0.6795$ | $0.9147$ | $0.4369$ | $0.0449$ | $94.72$ | $94.37$ | $94.23$ | $93.30$ | |

80 | $0.0232$ | $0.0004$ | $0.0004$ | $-0.0108$ | $0.5526$ | $0.7250$ | $0.3388$ | $0.0332$ | $94.25$ | $94.25$ | $94.35$ | $93.80$ | |

110 | $0.0115$ | $0.0064$ | $-0.0000$ | $-0.0080$ | $0.4558$ | $0.6124$ | $0.2883$ | $0.0273$ | $94.44$ | $94.48$ | $94.44$ | $94.00$ | |

140 | $0.0063$ | $0.0116$ | $0.0022$ | $-0.0064$ | $0.3903$ | $0.5348$ | $0.2473$ | $0.0237$ | $95.09$ | $94.61$ | $94.63$ | $94.57$ | |

170 | $0.0044$ | $0.0115$ | $0.0012$ | $-0.0052$ | $0.3343$ | $0.4843$ | $0.2265$ | $0.0213$ | $95.02$ | $94.72$ | $94.85$ | $94.91$ | |

200 | $0.0046$ | $0.0098$ | $0.0015$ | $-0.0045$ | $0.3087$ | $0.4471$ | $0.2112$ | $0.0193$ | $95.02$ | $94.52$ | $95.05$ | $95.26$ | |

0.75 | 20 | $-0.1173$ | $-0.0036$ | $0.0504$ | $-0.0487$ | $1.2419$ | $1.5266$ | $0.9314$ | $0.0898$ | $92.06$ | $92.28$ | $92.50$ | $88.70$ |

50 | $-0.0301$ | $0.0056$ | $-0.0034$ | $-0.0170$ | $0.7061$ | $0.9127$ | $0.4372$ | $0.0447$ | $94.23$ | $94.33$ | $94.17$ | $93.21$ | |

80 | $-0.0154$ | $0.0005$ | $0.0019$ | $-0.0102$ | $0.5709$ | $0.7250$ | $0.3387$ | $0.0332$ | $94.37$ | $94.25$ | $94.41$ | $93.69$ | |

110 | $-0.0161$ | $0.0070$ | $0.0021$ | $-0.0073$ | $0.4750$ | $0.6124$ | $0.2890$ | $0.0273$ | $94.65$ | $94.63$ | $94.41$ | $93.87$ | |

140 | $-0.0142$ | $0.0123$ | $0.0025$ | $-0.0058$ | $0.4074$ | $0.5350$ | $0.2472$ | $0.0237$ | $94.93$ | $94.77$ | $94.79$ | $94.49$ | |

170 | $-0.0122$ | $0.0132$ | $0.0017$ | $-0.0046$ | $0.3517$ | $0.4853$ | $0.2263$ | $0.0213$ | $94.84$ | $94.64$ | $94.90$ | $94.80$ | |

200 | $-0.0092$ | $0.0100$ | $0.0027$ | $-0.0040$ | $0.3230$ | $0.4468$ | $0.2117$ | $0.0193$ | $95.16$ | $94.60$ | $95.08$ | $95.20$ | |

0.90 | 20 | $-0.2778$ | $-0.0053$ | $0.0474$ | $-0.0487$ | $1.3766$ | $1.5197$ | $0.9277$ | $0.0898$ | $90.62$ | $92.62$ | $92.68$ | $88.56$ |

50 | $-0.0841$ | $0.0027$ | $-0.0043$ | $-0.0169$ | $0.7831$ | $0.9122$ | $0.4368$ | $0.0447$ | $93.66$ | $94.34$ | $94.12$ | $93.12$ | |

80 | $-0.0476$ | $-0.0014$ | $0.0014$ | $-0.0101$ | $0.6264$ | $0.7251$ | $0.3386$ | $0.0332$ | $93.82$ | $94.26$ | $94.42$ | $93.66$ | |

110 | $-0.0393$ | $0.0057$ | $0.0015$ | $-0.0072$ | $0.5235$ | $0.6127$ | $0.2890$ | $0.0273$ | $94.32$ | $94.78$ | $94.46$ | $93.86$ | |

140 | $-0.0325$ | $0.0111$ | $0.0020$ | $-0.0057$ | $0.4515$ | $0.5350$ | $0.2472$ | $0.0237$ | $94.56$ | $94.72$ | $94.82$ | $94.52$ | |

170 | $-0.0266$ | $0.0124$ | $0.0015$ | $-0.0046$ | $0.3933$ | $0.4853$ | $0.2265$ | $0.0213$ | $94.40$ | $94.70$ | $94.84$ | $94.78$ | |

200 | $-0.0218$ | $0.0094$ | $0.0025$ | $-0.0039$ | $0.3597$ | $0.4469$ | $0.2119$ | $0.0193$ | $94.94$ | $94.48$ | $95.22$ | $95.18$ |

ML Estimate | SE | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\tau}$ Levels | $\mathit{\tau}$ Levels | ||||||||||

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

VASI | ${\delta}_{0}$ | $-0.8467$ | $-0.6696$ | $-0.4757$ | $-0.2842$ | $-0.1132$ | $0.0481$ | $0.0455$ | $0.0441$ | $0.0442$ | $0.0456$ |

${\delta}_{1}$ | $0.0047$ | $0.0046$ | $0.0045$ | $0.0044$ | $0.0044$ | $0.0012$ | $0.0012$ | $0.0011$ | $0.0011$ | $0.0011$ | |

${\delta}_{2}$ | $0.0924$ | $0.0907$ | $0.0891$ | $0.0878$ | $0.0869$ | $0.0069$ | $0.0067$ | $0.0066$ | $0.0065$ | $0.0064$ | |

${\delta}_{3}$ | $-0.9579$ | $-0.9386$ | $-0.9203$ | $-0.9052$ | $-0.8944$ | $0.0380$ | $0.0372$ | $0.0364$ | $0.0357$ | $0.0352$ | |

${\delta}_{4}$ | $-0.1205$ | $-0.1188$ | $-0.1172$ | $-0.1161$ | $-0.1154$ | $0.0539$ | $0.0529$ | $0.0519$ | $0.0512$ | $0.0506$ | |

${\delta}_{5}$ | $-0.2657$ | $-0.2606$ | $-0.2557$ | $-0.2517$ | $-0.2488$ | $0.0520$ | $0.0509$ | $0.0500$ | $0.0492$ | $0.0487$ | |

$\theta $ | $0.0302$ | $0.0301$ | $0.0301$ | $0.0300$ | $0.0299$ | $0.0024$ | $0.0024$ | $0.0024$ | $0.0024$ | $0.0024$ | |

ULOG | ${\delta}_{0}$ | $-0.8528$ | $-0.6641$ | $-0.4753$ | $-0.2866$ | $-0.0979$ | $0.0468$ | $0.0441$ | $0.0432$ | $0.0443$ | $0.0471$ |

${\delta}_{1}$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0011$ | $0.0011$ | $0.0011$ | $0.0011$ | $0.0011$ | |

${\delta}_{2}$ | $0.0888$ | $0.0888$ | $0.0888$ | $0.0888$ | $0.0888$ | $0.0066$ | $0.0066$ | $0.0066$ | $0.0066$ | $0.0066$ | |

${\delta}_{3}$ | $-0.9324$ | $-0.9324$ | $-0.9324$ | $-0.9324$ | $-0.9324$ | $0.0359$ | $0.0359$ | $0.0359$ | $0.0359$ | $0.0359$ | |

${\delta}_{4}$ | $-0.1223$ | $-0.1223$ | $-0.1223$ | $-0.1223$ | $-0.1223$ | $0.0513$ | $0.0513$ | $0.0513$ | $0.0513$ | $0.0513$ | |

${\delta}_{5}$ | $-0.2385$ | $-0.2385$ | $-0.2385$ | $-0.2385$ | $-0.2385$ | $0.0486$ | $0.0486$ | $0.0486$ | $0.0486$ | $0.0486$ | |

$\theta $ | $5.8208$ | $5.8208$ | $5.8208$ | $5.8208$ | $5.8208$ | $0.2824$ | $0.2824$ | $0.2824$ | $0.2824$ | $0.2824$ | |

JOSB | ${\delta}_{0}$ | $-0.8661$ | $-0.6784$ | $-0.4699$ | $-0.2614$ | $-0.0738$ | $0.0494$ | $0.0474$ | $0.0467$ | $0.0474$ | $0.0494$ |

${\delta}_{1}$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0047$ | $0.0012$ | $0.0012$ | $0.0012$ | $0.0012$ | $0.0012$ | |

${\delta}_{2}$ | $0.0916$ | $0.0916$ | $0.0916$ | $0.0916$ | $0.0916$ | $0.0068$ | $0.0068$ | $0.0068$ | $0.0068$ | $0.0068$ | |

${\delta}_{3}$ | $-0.9379$ | $-0.9379$ | $-0.9379$ | $-0.9379$ | $-0.9379$ | $0.0373$ | $0.0373$ | $0.0373$ | $0.0373$ | $0.0373$ | |

${\delta}_{4}$ | $-0.1174$ | $-0.1174$ | $-0.1174$ | $-0.1174$ | $-0.1174$ | $0.0546$ | $0.0546$ | $0.0546$ | $0.0546$ | $0.0546$ | |

${\delta}_{5}$ | $-0.2635$ | $-0.2635$ | $-0.2635$ | $-0.2635$ | $-0.2635$ | $0.0524$ | $0.0524$ | $0.0524$ | $0.0524$ | $0.0524$ | |

$\theta $ | $3.2351$ | $3.2351$ | $3.2351$ | $3.2351$ | $3.2351$ | $0.1325$ | $0.1325$ | $0.1325$ | $0.1325$ | $0.1325$ | |

KUMA | ${\delta}_{0}$ | $-1.1035$ | $-0.8088$ | $-0.5298$ | $-0.2908$ | $-0.1018$ | $0.0515$ | $0.0484$ | $0.0486$ | $0.0516$ | $0.0559$ |

${\delta}_{1}$ | $0.0036$ | $0.0038$ | $0.0042$ | $0.0045$ | $0.0048$ | $0.0009$ | $0.0010$ | $0.0010$ | $0.0011$ | $0.0011$ | |

${\delta}_{2}$ | $0.0729$ | $0.0772$ | $0.0823$ | $0.0874$ | $0.0921$ | $0.0055$ | $0.0058$ | $0.0061$ | $0.0064$ | $0.0068$ | |

${\delta}_{3}$ | $-0.7302$ | $-0.7715$ | $-0.8201$ | $-0.8699$ | $-0.9150$ | $0.0328$ | $0.0345$ | $0.0367$ | $0.0391$ | $0.0414$ | |

${\delta}_{4}$ | $-0.0724$ | $-0.0744$ | $-0.0763$ | $-0.0776$ | $-0.0783$ | $0.0475$ | $0.0503$ | $0.0537$ | $0.0572$ | $0.0604$ | |

${\delta}_{5}$ | $-0.1963$ | $-0.2057$ | $-0.2163$ | $-0.2267$ | $-0.2359$ | $0.0467$ | $0.0496$ | $0.0530$ | $0.0565$ | $0.0597$ | |

$\theta $ | $4.7048$ | $4.7148$ | $4.7237$ | $4.7307$ | $4.7360$ | $0.2103$ | $0.2103$ | $0.2101$ | $0.2100$ | $0.2099$ | |

UWEI | ${\delta}_{0}$ | $-0.8251$ | $-0.6813$ | $-0.4940$ | $-0.2697$ | $-0.0288$ | $0.0464$ | $0.0440$ | $0.0423$ | $0.0422$ | $0.0441$ |

${\delta}_{1}$ | $0.0058$ | $0.0055$ | $0.0051$ | $0.0047$ | $0.0044$ | $0.0014$ | $0.0014$ | $0.0013$ | $0.0012$ | $0.0011$ | |

${\delta}_{2}$ | $0.0859$ | $0.0820$ | $0.0773$ | $0.0722$ | $0.0674$ | $0.0073$ | $0.0069$ | $0.0065$ | $0.0060$ | $0.0056$ | |

${\delta}_{3}$ | $-0.9658$ | $-0.9193$ | $-0.8632$ | $-0.8028$ | $-0.7460$ | $0.0426$ | $0.0402$ | $0.0374$ | $0.0346$ | $0.0321$ | |

${\delta}_{4}$ | $-0.1312$ | $-0.1266$ | $-0.1210$ | $-0.1149$ | $-0.1089$ | $0.0551$ | $0.0527$ | $0.0498$ | $0.0467$ | $0.0437$ | |

${\delta}_{5}$ | $-0.3551$ | $-0.3389$ | $-0.3192$ | $-0.2978$ | $-0.2774$ | $0.0554$ | $0.0529$ | $0.0499$ | $0.0466$ | $0.0434$ | |

$\theta $ | $5.9760$ | $5.9797$ | $5.9849$ | $5.9918$ | $6.0001$ | $0.2430$ | $0.2429$ | $0.2428$ | $0.2427$ | $0.2427$ |

$\mathit{\tau}$ Levels | |||||
---|---|---|---|---|---|

Model | 0.10 | 0.25 | 0.50 | 0.75 | 0.90 |

VASI | $-894.6886$ | $-895.7425$ | $-896.9891$ | $-898.2973$ | $-899.5114$ |

ULOG | $-888.4015$ | $-888.4015$ | $-888.4015$ | $-888.4015$ | $-888.4015$ |

JOSB | $-883.9673$ | $-883.9673$ | $-883.9673$ | $-883.9673$ | $-883.9673$ |

KUMA | $-839.6604$ | $-842.2240$ | $-845.0303$ | $-847.6574$ | $-849.8131$ |

UWEI | $-838.1505$ | $-839.3637$ | $-840.9321$ | $-842.7567$ | $-844.6025$ |

Distribution | ||||
---|---|---|---|---|

State | Parameter | VASI | Beta | Simplex |

Minas Gerais | $\theta $ | $-2.3233$ | $-1.1733$ | $0.0759$ |

$\left(0.0487\right)$ | $\left(0.0293\right)$ | $\left(0.0244\right)$ | ||

${\delta}_{0}$ | $0.4104$ | $0.4079$ | $0.4101$ | |

$\left(0.0180\right)$ | $\left(0.0181\right)$ | $\left(0.0178\right)$ | ||

${\delta}_{1}$ | $-5.7168$ | $-5.6945$ | $-5.8076$ | |

$\left(0.3359\right)$ | $\left(0.3403\right)$ | $\left(0.3251\right)$ | ||

AIC | $-1252.4860$ | $-1248.1210$ | $-1250.7680$ | |

Piauí | $\theta $ | $-2.9806$ | $-1.7376$ | $0.0987$ |

$\left(0.0949\right)$ | $\left(0.0543\right)$ | $\left(0.0475\right)$ | ||

${\delta}_{0}$ | $1.3165$ | $1.3099$ | $1.3468$ | |

$\left(0.0562\right)$ | $\left(0.0544\right)$ | $\left(0.0597\right)$ | ||

${\delta}_{1}$ | $-2.9229$ | $-3.0062$ | $-2.5073$ | |

$\left(0.6604\right)$ | $\left(0.6511\right)$ | $\left(0.6691\right)$ | ||

AIC | $-646.7614$ | $-646.3025$ | $-643.6276$ |

Distribution | |||
---|---|---|---|

Parameter | VASI | Beta | Simplex |

${\delta}_{0}$ | $0.3986$ $\left(0.0178\right)$ | $0.3954$ $\left(0.0178\right)$ | $0.3985$ $\left(0.0178\right)$ |

${\delta}_{1}$ | $0.7499$ $\left(0.0433\right)$ | $0.7617$ $\left(0.0423\right)$ | $0.7224$ $\left(0.0461\right)$ |

${\delta}_{2}$ | $-5.1550$ $\left(0.3038\right)$ | $-5.1200$ $\left(0.3032\right)$ | $-5.2187$ $\left(0.3057\right)$ |

${\beta}_{0}$ | $-2.3173$ $\left(0.0489\right)$ | $-1.1689$ $\left(0.0294\right)$ | $0.0778$ $\left(0.0244\right)$ |

${\beta}_{1}$ | $0.6194$ $\left(0.1080\right)$ | $-0.5481$ $\left(0.0625\right)$ | $0.0556$ $\left(0.0542\right)$ |

AIC | $-1887.1730$ | $-1882.9680$ | $-1877.7550$ |

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## Share and Cite

**MDPI and ACS Style**

Mazucheli, J.; Alves, B.; Korkmaz, M.Ç.; Leiva, V.
Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications. *Mathematics* **2022**, *10*, 1389.
https://doi.org/10.3390/math10091389

**AMA Style**

Mazucheli J, Alves B, Korkmaz MÇ, Leiva V.
Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications. *Mathematics*. 2022; 10(9):1389.
https://doi.org/10.3390/math10091389

**Chicago/Turabian Style**

Mazucheli, Josmar, Bruna Alves, Mustafa Ç. Korkmaz, and Víctor Leiva.
2022. "Vasicek Quantile and Mean Regression Models for Bounded Data: New Formulation, Mathematical Derivations, and Numerical Applications" *Mathematics* 10, no. 9: 1389.
https://doi.org/10.3390/math10091389