Title:	Parametric Quantile Regression Models for Bounded Data
Version:	0.0.6
Maintainer:	André F. B. Menezes <andrefelipemaringa@gmail.com>
Description:	A collection of parametric quantile regression models for bounded data. At present, the package provides 13 parametric quantile regression models. It can specify regression structure for any quantile and shape parameters. It also provides several S3 methods to extract information from fitted model, such as residual analysis, prediction, plotting, and model comparison. For more computation efficient the [dpqr]'s, likelihood, score and hessian functions are written in C++. For further details see Mazucheli et. al (2022) <doi:10.1016/j.cmpb.2022.106816>.
License:	Apache License (≥ 2)
Encoding:	UTF-8
ByteCompile:	yes
LazyData:	true
LinkingTo:	Rcpp
Imports:	Rcpp, optimx, stats, quantreg, Formula, MASS, numDeriv
Suggests:	testthat (≥ 3.0.0), rmarkdown, knitr, lmtest, ggplot2, covr
Depends:	R (≥ 3.5.0)
RoxygenNote:	7.2.1
NeedsCompilation:	yes
URL:	https://andrmenezes.github.io/unitquantreg/
BugReports:	https://github.com/AndrMenezes/unitquantreg/issues
VignetteBuilder:	knitr
Packaged:	2023-09-05 09:00:00 UTC; amenezes
Author:	André F. B. Menezes [aut, cre], Josmar Mazucheli [aut]
Repository:	CRAN
Date/Publication:	2023-09-06 09:10:02 UTC

Overview of the unitquantreg package

Description

The unitquantreg R package provides a collection of parametric quantile regression models for bounded data. At present, the package provides 13 parametric quantile regression models. It also enables several S3 methods to extract information from fitted model, such as residual analysis, prediction, plotting, and model comparison.

Author(s)

André F. B. Menezes andrefelipemaringa@gmail.com

Josmar Mazucheli jmazucheli@gmail.com

The arcsecant hyperbolic Weibull distribution

Description

Density function, distribution function, quantile function and random number generation function for the arcsecant hyperbolic Weibull distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dashw(x, mu, theta, tau = 0.5, log = FALSE)

pashw(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qashw(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rashw(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y;\alpha, \theta)=\frac{\alpha \theta}{y\sqrt{1-y^2}} \mathrm{arcsech}(y)^{\theta-1}\exp\left [ -\alpha \mathrm{arcsech}(y)^\theta \right ]

Cumulative distribution function

F(y;\alpha, \theta)=\exp\left [ -\alpha \mathrm{arcsech}(y)^\theta \right ]

Quantile function

Q(\tau;\alpha, \theta)= \mathrm{sech}\left \{ \left [ -\alpha^{-1} \log(\tau)\right ]^{\frac{1}{\theta}} \right \}

Reparameterization

\alpha = g^{-1}(\mu) = -\frac{\log(\tau)}{\mathrm{arcsech}(\mu)^\theta}

where \theta >0 is the shape parameter and \mathrm{arcsech}(y)= \log\left[\left( 1+\sqrt{1-y^2} \right)/y \right].

Value

dashw gives the density, pashw gives the distribution function, qashw gives the quantile function and rashw generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli

André F. B. Menezes

References

Korkmaz, M. C., Chesneau, C. and Korkmaz, Z. S., (2021). A new alternative quantile regression model for the bounded response with educational measurements applications of OECD countries. Journal of Applied Statistics, 1–25.

Examples

set.seed(6969)
x <- rashw(n = 1000, mu = 0.5, theta = 2.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1L], to = R[2L], by = 0.01)
hist(x, prob = TRUE, main = 'arcsecant hyperbolic Weibull')
lines(S, dashw(x = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pashw(q = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qashw(p = S, mu = 0.5, theta = 2.5, tau = 0.5), col = 2)

Percentage of body fat data set

Description

The body fat percentage of individuals assisted in a public hospital in Curitiba, Paraná, Brasil.

Usage

data(bodyfat, package = "unitquantreg")

Format

A data.frame with 298 observations and 9 columns:

• arms: Arms fat percentage.

• legs: Legs fat percentage.

• body: Body fat percentage.

• android: Android fat percentage.

• gynecoid: Ginecoid fat percentage.

• bmi: Body mass index - 24.71577.

• age: Age - 46.00.

• sex: Sex of individual. Female or male.

• ipaq: Factor variable indicating the sedentary, insufficiently active or active.

Author(s)

André F. B. Menezes

Josmar Mazucheli

Source

http://www.leg.ufpr.br/doku.php/publications:papercompanions:multquasibeta

References

Petterle, R. R., Bonat, W. H., Scarpin, C. T., Jonasson, T., and Borba, V. Z. C., (2020). Multivariate quasi-beta regression models for continuous bounded data. The International Journal of Biostatistics, 1–15, (preprint).

Mazucheli, J., Leiva, V., Alves, B., and Menezes A. F. B., (2021). A new quantile regression for modeling bounded data under a unit Birnbaum-Saunders distribution with applications in medicine and politics. Symmetry, 13(4) 1–21.

(Half-)Normal probability plots with simulated envelopes for unitquantreg objects

Description

Produces a (half-)normal probability plot from a fitted model object of class unitquantreg.

Usage

hnp(object, ...)

## S3 method for class 'unitquantreg'
hnp(
  object,
  nsim = 99,
  halfnormal = TRUE,
  plot = TRUE,
  output = TRUE,
  level = 0.95,
  resid.type = c("quantile", "cox-snell"),
  ...
)

Arguments

Details

Residuals plots with simulated envelope were proposed by Atkinson (1981) and can be construct as follows:

1. generate sample set of n independent observations from the estimated parameters of the fitted model;

2. fit the model using the generated sample, if halfnormal is TRUE compute the absolute values of the residuals and arrange them in order;

3. repeat steps (1) and (2) nsim number of times;

4. consider the n sets of the nsim ordered statistics of the residuals, then for each set compute the quantile level/2, the median and the quantile 1 - level/2;

5. plot these values and the ordered residuals of the original sample set versus the expected order statistics of a (half)-normal distribution, which is approximated as

   G^{-1} \left(\frac{i + n - 0.125}{2n + 0.5} \right)

   for half-normal plots, i.e., halfnormal=TRUE or

   G^{-1} \left(\frac{i - 0.375}{n + 0.25}\right)

   for normal plots, i.e., halfnormal=FALSE, where G(\cdot) is the the cumulative distribution function of standard Normal distribution for quantile residuals or the standard exponential distribution for the cox-snell residuals.

According to Atkinson (1981), if the model was correctly specified then no more than level100% of the observations are expected to appear outside the envelope bands. Additionally, if a large proportion of the observations lies outside the envelope, thus one has evidence against the adequacy of the fitted model.

Value

A list with the following components in ordered (and absolute if halfnormal is TRUE) values:

Author(s)

André F. B. Menezes

References

Atkinson, A. C., (1981). Two graphical displays for outlying and influential observations in regression. Biometrika 68(1), 13–20.

See Also

residuals.unitquantreg

The Johnson SB distribution

Description

Density function, distribution function, quantile function and random number generation function for the Johnson SB distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

djohnsonsb(x, mu, theta, tau = 0.5, log = FALSE)

pjohnsonsb(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qjohnsonsb(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rjohnsonsb(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\theta }{\sqrt{2\pi }}\frac{1}{y(1-y)}\exp\left\{ -\frac{1}{2}\left[\alpha +\theta \log\left(\frac{y}{1-y}\right)\right] ^{2}\right\}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\Phi \left[ \alpha +\theta \log \left( \frac{y}{1-y}\right) \right]

Quantile function

Q(\tau \mid \alpha ,\theta )=\frac{\exp \left[ \frac{\Phi ^{-1}(\tau)-\alpha }{\theta }\right] }{1+\exp \left[ \frac{\Phi ^{-1}(\tau )-\alpha }{\theta }\right] }

Reparameterization

\alpha =g^{-1}(\mu )=\Phi ^{-1}(\tau )-\theta \log \left( \frac{\mu }{1-\mu }\right)

Value

djohnsonsb gives the density, pjohnsonsb gives the distribution function, qjohnsonsb gives the quantile function and rjohnsonsb generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli

André F. B. Menezes

References

Lemonte, A. J. and Bazán, J. L., (2015). New class of Johnson SB distributions and its associated regression model for rates and proportions. Biometrical Journal, 58(4), 727–746.

Johnson, N. L., (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36(1), 149–176.

Examples

set.seed(123)
x <- rjohnsonsb(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'Johnson SB')
lines(S, djohnsonsb(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pjohnsonsb(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qjohnsonsb(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The Kumaraswamy distribution

Description

Density function, distribution function, quantile function and random number generation for the Kumaraswamy distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dkum(x, mu, theta, tau = 0.5, log = FALSE)

pkum(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qkum(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rkum(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\alpha \theta y^{\theta -1}(1-y^{\theta })^{\alpha-1}

Cumulative distribution function

F(y\mid \alpha ,\theta )=1-\left( 1-y^{\theta }\right) ^{\alpha }

Quantile function

Q(\tau \mid \alpha ,\theta )=\left[ 1-\left( 1-\tau \right) ^{\frac{1}{\alpha }}\right] ^{\frac{1}{\theta }}

Reparameterization

\alpha=g^{-1}(\mu )=\frac{\log (1-\tau )}{\log (1-\mu ^{\theta })}

Value

dkum gives the density, pkum gives the distribution function, qkum gives the quantile function and rkum generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli

André F. B. Menezes

References

Kumaraswamy, P., (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology, 46(1), 79–88.

Jones, M. C., (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology, 6(1), 70-81.

Examples

set.seed(123)
x <- rkum(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'Kumaraswamy')
lines(S, dkum(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pkum(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qkum(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The Log-extended exponential-geometric distribution

Description

Density function, distribution function, quantile function and random number generation function for the Log-extended exponential-geometric distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dleeg(x, mu, theta, tau = 0.5, log = FALSE)

pleeg(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qleeg(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rleeg(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\theta \left( 1+\alpha \right) y^{\theta -1}}{\left( 1+\alpha y^{\theta }\right) ^{2}}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\frac{\left( 1+\alpha \right) y^{\theta }}{1+\alpha y^{\theta }}

Quantile function

Q(\tau \mid \alpha ,\theta )=\left[ \frac{\tau }{1+\alpha \left( 1-\tau\right) }\right] ^{\frac{1}{\theta }}

Reparameterization

\alpha=g^{-1}(\mu )=-\frac{1-\tau \mu ^{\theta }}{\left( 1-\tau \right) }

Value

dleeg gives the density, pleeg gives the distribution function, qleeg gives the quantile function and rleeg generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Jodrá, P. and Jiménez-Gamero, M. D., (2020). A quantile regression model for bounded responses based on the exponential-geometric distribution. Revstat - Statistical Journal, 18(4), 415–436.

Examples

set.seed(123)
x <- rleeg(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'Log-extended exponential-geometric')
lines(S, dleeg(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pleeg(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qleeg(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

Likelihood-based statistics of fit for unitquantreg objects.

Description

Computes the likelihood-based statistics (Neg2LogLike, AIC, BIC and HQIC) from unitquantreg objects.

Usage

likelihood_stats(..., lt = NULL)

## S3 method for class 'likelihood_stats'
print(x, ...)

Arguments

Details

Neg2LogLike: The log-likelihood is reported as

Neg2LogLike= -2\log(L)

AIC: The Akaike's information criterion (AIC) is defined as

AIC = -2\log(L)+2p

BIC: The Schwarz Bayesian information criterion (BIC) is defined as

BIC = -2\log(L) + p\log(n)

HQIC: The Hannan and Quinn information criterion (HQIC) is defined as

HQIC = -2\log(L) + 2p\log[\log(n)]

where L is the likelihood function.

Value

A list with class "likelihood_stats" containing the following components:

Author(s)

André F. B. Menezes

Josmar Mazucheli

References

Akaike, H. (1974). A new look at the statistical model identification. IEEE Transaction on Automatic Control, 19(6), 716–723.

Hannan, E. J. and Quinn, B. G. (1979). The determination of the order of an autoregression. Journal of the Royal Statistical Society, Series B, 41(2), 190–195.

Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461–464.

Examples

data(sim_bounded, package = "unitquantreg")
sim_bounded_curr <- sim_bounded[sim_bounded$family == "uweibull", ]

models <- c("uweibull", "kum", "ulogistic")
lt_fits <- lapply(models, function(fam) {
  unitquantreg(formula = y1 ~ x, tau = 0.5, data = sim_bounded_curr,
               family = fam)
})

ans <- likelihood_stats(lt = lt_fits)
ans

Log-likelihood, score vector and hessian matrix.

Description

Internal functions using in unitquantreg.fit to compute the negative log-likelihood function, the score vector and the hessian matrix using analytic expressions written in C++.

Usage

loglike_unitquantreg(par, tau, family, linkobj, linkobj.theta, X, Z, y)

Arguments

Methods for unitquantreg and unitquantregs objects

Description

Methods for extracting information from fitted regression models objects of class unitquantreg and unitquantregs.

Usage

## S3 method for class 'unitquantreg'
print(x, digits = max(4, getOption("digits") - 3), ...)

## S3 method for class 'unitquantreg'
summary(object, correlation = FALSE, ...)

## S3 method for class 'unitquantreg'
coef(object, type = c("full", "quantile", "shape"), ...)

## S3 method for class 'unitquantreg'
vcov(object, ...)

## S3 method for class 'unitquantreg'
logLik(object, ...)

## S3 method for class 'unitquantreg'
confint(object, parm, level = 0.95, ...)

## S3 method for class 'unitquantreg'
fitted(object, type = c("quantile", "shape", "full"), ...)

## S3 method for class 'unitquantreg'
terms(x, type = c("quantile", "shape"), ...)

## S3 method for class 'unitquantreg'
model.frame(formula, ...)

## S3 method for class 'unitquantreg'
model.matrix(object, type = c("quantile", "shape"), ...)

## S3 method for class 'unitquantreg'
update(object, formula., ..., evaluate = TRUE)

## S3 method for class 'unitquantregs'
print(x, digits = max(3, getOption("digits") - 3), ...)

## S3 method for class 'unitquantregs'
summary(object, digits = max(3, getOption("digits") - 3), ...)

Arguments

Value

The summary method gives Wald tests for the regressions coefficients based on the observed Fisher information matrix. As usual the summary method returns a list with relevant model statistics and estimates, which can be printed using the print method.

The coef, vcov, confint and fitted methods can be use to extract, respectively, the estimated coefficients, the estimated covariance matrix, the Wald confidence intervals, and fitted values.

A logLik method is also provide, then the AIC function can be use to calculated the Akaike Information Criterion.

The generic methods terms, model.frame, model.matrix, update and are also provided.

Author(s)

André F. B. Menezes

Examples

data(sim_bounded, package = "unitquantreg")
sim_bounded_curr <- sim_bounded[sim_bounded$family == "uweibull", ]
fit_1 <- unitquantreg(formula = y1 ~ x + z + I(x^2) | z + x,
                      data = sim_bounded_curr,
                      family = "uweibull",
                      tau = 0.5, link.theta = "log")
fit_1
summary(fit_1)
vcov(fit_1)
coef(fit_1)
confint(fit_1)
terms(fit_1)
model.frame(fit_1)[1:5, ]
model.matrix(fit_1)[1:5, ]
update(fit_1, . ~ . -x)
update(fit_1, . ~ . -z)
update(fit_1, . ~ . -I(x^2))
update(fit_1, . ~ . | . -z)
update(fit_1, . ~ . | . -x)

Pairwise vuong test

Description

Calculate pairwise comparisons between fitted models performing vuong test for objects of class unitquantreg.

Usage

pairwise.vuong.test(
  ...,
  lt,
  p.adjust.method = p.adjust.methods,
  alternative = c("two.sided", "less", "greater")
)

Arguments

Value

Object of class "pairwise.htest"

See Also

vuong.test, p.adjust

Examples

data(sim_bounded, package = "unitquantreg")
sim_bounded_curr <- sim_bounded[sim_bounded$family == "uweibull", ]

models <- c("uweibull", "kum", "ulogistic")
lt_fits <- lapply(models, function(fam) {
  unitquantreg(formula = y1 ~ x, tau = 0.5, data = sim_bounded_curr,
               family = fam)
})

ans <- pairwise.vuong.test(lt = lt_fits)
ans

Plot method for unitquantreg objects

Description

Provide diagnostic plots to check model assumptions for fitted model of class unitquantreg.

Usage

## S3 method for class 'unitquantreg'
plot(
  x,
  which = 1L:4L,
  caption = c("Residuals vs. indices of obs.", "Residuals vs. linear predictor",
    "Working response vs. linear predictor", "Half-normal plot of residuals"),
  sub.caption = paste(deparse(x$call), collapse = "\n"),
  main = "",
  ask = prod(par("mfcol")) < length(which) && dev.interactive(),
  ...,
  add.smooth = getOption("add.smooth"),
  type = "quantile",
  nsim = 99L,
  level = 0.95
)

Arguments

Details

The plot method for unitquantreg objects produces four types of diagnostic plot.

The which argument can be used to select a subset of currently four supported plot, which are: Residuals versus indices of observations (which = 1); Residuals versus linear predictor (which = 2); Working response versus linear predictor (which = 3) to check possible misspecification of link function; Half-normal plot of residuals (which = 4) to check distribution assumption.

Value

No return value, called for side effects.

Author(s)

André F. B. Menezes

References

Dunn, P. K. and Smyth, G. K. (2018) Generalized Linear Models With Examples in R, Springer, New York.

See Also

residuals.unitquantreg, hnp.unitquantreg, unitquantreg.

Plot method for unitquantregs objects

Description

Provide two type of plots for unitquantregs objects.

Usage

## S3 method for class 'unitquantregs'
plot(
  x,
  which = c("coef", "conddist"),
  output_df = FALSE,
  parm = NULL,
  level = 0.95,
  mean_effect = FALSE,
  mfrow = NULL,
  mar = NULL,
  ylim = NULL,
  main = NULL,
  col = gray(c(0, 0.75)),
  border = NULL,
  cex = 1,
  pch = 20,
  type = "b",
  xlab = bquote("Quantile level (" * tau * ")"),
  ylab = "Estimate effect",
  dist_type = c("density", "cdf"),
  at_avg = TRUE,
  at_obs = NULL,
  legend_position = "topleft",
  ...
)

Arguments

Details

The plot method for unitquantregs objects is inspired in PROC QUANTREG of SAS/STAT. This plot method provide two type of visualizations.

If which = "coef" plot the estimated coefficients for several quantiles.

If which = "conddist" plot the conditional distribution at specific values of covariates. The conditional distribution could be the cumulative distribution function if dist_type = "cdf" or the probability density function if dist_type = "pdf".

Value

If output_df = TRUE then returns a data.frame used to plot. Otherwise, no return value, called for side effects.

Author(s)

André F. B. Menezes

See Also

plot.unitquantreg.

Prediction method for unitquantreg class

Description

Extract various types of predictions from unit quantile regression models.

Usage

## S3 method for class 'unitquantreg'
predict(
  object,
  newdata,
  type = c("link", "quantile", "shape", "terms"),
  interval = c("none", "confidence"),
  level = 0.95,
  se.fit = FALSE,
  ...
)

Arguments

Value

If se.fit = FALSE then returns a data.frame with predict values and confidence interval if interval = TRUE.

If se.fit = TRUE returns a list with components:

For type = "terms" the output is a data.frame with a columns per term.

Author(s)

André F. B. Menezes

Residuals method for unitquantreg objects

Description

Extract various types of residuals from unit quantile regression models.

Usage

## S3 method for class 'unitquantreg'
residuals(object, type = c("quantile", "cox-snell", "working", "partial"), ...)

Arguments

Details

The residuals method can compute quantile and Cox-Snell residuals. These residuals are defined, respectively, by

r_{Q} = \Phi^{-1}\left[ F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]

and

r_{CS} = -\log\left[1- F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]

where \widehat{\mu}_i and \widehat{\theta}_i are the fitted values of parameters \mu and \theta, F(\cdot \mid \cdot, \cdot) is the cumulative distribution function (c.d.f.) and \Phi(\cdot) is the c.d.f. of standard Normal distribution.

Apart from the variability due the estimates of parameters,if the fitted regression model is correctly specified then the quantile residuals, r_Q, follow a standard Normal distribution and the Cox-Snell residuals, r_{CS}, follow a standard exponential distribution.

Value

Numeric vector of residuals extract from an object of class unitquantreg.

Author(s)

André F. B. Menezes

References

Cox, D. R. and Snell E. J., (1968). A general definition of residuals. Journal of the Royal Statistical Society - Series B, 30(2), 248–265.

Dunn, P. K. and Smyth, G. K., (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5(3), 236–244.

Simulated data set

Description

This data set was simulated from all families of distributions available in unitquantreg package considering the median, i.e., \tau=0.5.

Usage

data(sim_bounded, package = "unitquantreg")

Format

data.frame with 1300 observations and 5 columns:

• y1: simulated response variable with constant shape parameter, \theta = 2.

• y2: simulated response variable with regression structure in the shape parameter, \theta_i = \exp(\zeta_i), where \zeta_i = \mathbf{z}_i^\top\,\boldsymbol{\gamma}.

• x: covariate related to \mu_i, i.e., the median.

• z: covariate related to \theta_i, i.e., the shape parameter.

• family: string indicating the family of distribution.

Details

There are two response variable, namely y1 and y2. The former was simulated considering a regression structure for \mu and one covariate simulated from a standard uniform distribution, where the true vector of coefficients for \mu is \boldsymbol{\beta} = (1, 2) and \theta = 2. The latter was simulated assuming a regression structure for both \mu and \theta (shape parameter) and only one independent covariates simulated from two standard uniform distributions. The true vectors of coefficients for \mu and \theta are \boldsymbol{\beta} = (1, 2) and \boldsymbol{\gamma} = (-1, 1), respectively.

Author(s)

André F. B. Menezes

The unit-Birnbaum-Saunders distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Birnbaum-Saunders distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dubs(x, mu, theta, tau = 0.5, log = FALSE)

pubs(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qubs(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rubs(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{1}{2y\alpha \theta \sqrt{2\pi }}\left[\left( -\frac{\alpha }{\log (y)}\right) ^{\frac{1}{2}}+\left( -\frac{\alpha}{\log (y)}\right) ^{\frac{3}{2}}\right] \exp \left[ \frac{1}{2\theta ^{2}}\left( 2+\frac{\log (y)}{\alpha }+\frac{\alpha }{\log (y)}\right) \right]

Cumulative distribution function

F(y\mid \alpha ,\theta )=1-\Phi \left\{ \frac{1}{\theta }\left[ \left( -\frac{\log (y)}{\alpha }\right) ^{\frac{1}{2}}-\left( -\frac{\alpha }{\log(y)}\right) ^{\frac{1}{2}}\right] \right\}

Quantile function

Q\left( \tau \mid \alpha ,\theta \right) ={\exp }\left\{ -{\frac{2\alpha}{2+\left[ {\theta }\Phi ^{-1}\left( 1-\tau \right) \right] ^{2}-{\theta } \Phi ^{-1}\left( 1-\tau \right) \sqrt{4+\left[ {\theta }\Phi ^{-1}\left(1-\tau \right) \right] ^{2}}}}\right\}

Reparameterization

\alpha=g^{-1}(\mu )=\log \left( \mu \right) g\left( \theta ,\tau \right)

where g\left( \theta ,\tau \right) =-\frac{1}{2}\left\{ 2+\left[ {\theta }\Phi^{-1}\left( 1-\tau \right) \right] ^{2}-{\theta }\Phi ^{-1}\left( 1-\tau\right) \sqrt{4+{\theta }\Phi ^{-1}\left( 1-\tau \right) }\right\} .

Value

dubs gives the density, pubs gives the distribution function, qubs gives the quantile function and rubs generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Birnbaum, Z. W. and Saunders, S. C., (1969). A new family of life distributions. Journal of Applied Probability, 6(2), 637–652. Mazucheli, J., Menezes, A. F. B. and Dey, S., (2018). The unit-Birnbaum-Saunders distribution with applications. Chilean Journal of Statistics, 9(1), 47–57.

Mazucheli, J., Alves, B. and Menezes, A. F. B., (2021). A new quantile regression for modeling bounded data under a unit Birnbaum-Saunders distribution with applications. Simmetry, (), 1–28.

Examples

set.seed(123)
x <- rubs(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Birnbaum-Saunders')
lines(S, dubs(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pubs(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qubs(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The unit-Burr-XII distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Burr-XII distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

duburrxii(x, mu, theta, tau = 0.5, log = FALSE)

puburrxii(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

quburrxii(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

ruburrxii(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha, \theta )=\frac{\alpha \theta }{y}\left[ -\log (y)\right]^{\theta -1}\left\{ 1+\left[ -\log (y)\right] ^{\theta }\right\} ^{-\alpha -1}

Cumulative distribution function

F(y\mid \alpha, \theta )=\left\{ 1+\left[ -\log (y)\right] ^{\theta}\right\} ^{-\alpha }

Quantile function

Q(\tau \mid \alpha, \theta )=\exp \left[ -\left( \tau ^{-\frac{1}{\alpha }}-1\right)^{\frac{1}{\theta }} \right]

Reparameterization

\alpha=g^{-1}(\mu)=\frac{\log\left ( \tau^{-1} \right )}{\log\left [ 1+\log\left ( \frac{1}{\mu} \right )^\theta \right ]}

Value

duburrxii gives the density, puburrxii gives the distribution function, quburrxii gives the quantile function and ruburrxii generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Korkmaz M. C. and Chesneau, C., (2021). On the unit Burr-XII distribution with the quantile regression modeling and applications. Computational and Applied Mathematics, 40(29), 1–26.

Examples

set.seed(123)
x <- ruburrxii(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Burr-XII')
lines(S, duburrxii(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, puburrxii(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(quburrxii(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The unit-Chen distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Chen distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

duchen(x, mu, theta, tau = 0.5, log = FALSE)

puchen(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

quchen(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

ruchen(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\alpha \theta }{y}\left[ -\log (y)\right]^{\theta -1}\exp \left\{ \left[ -\log \left( y\right) \right]^{\theta}\right\} \exp \left\{ \alpha \left\{ 1-\exp \left[ \left( -\log (y)\right)^{\theta }\right] \right\} \right\}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\exp \left\{ \alpha \left\{ 1-\exp \left[ \left(-\log (y)\right)^{\theta }\right] \right\} \right\}

Quantile function

Q\left( \tau \mid \alpha ,\theta \right) =\exp \left\{ -\left[ \log \left( 1-{\frac{\log \left( \tau \right) }{\alpha }}\right) \right]^{\frac{1}{\theta}}\right\}

Reparameterization

\alpha=g^{-1}(\mu )={\frac{\log \left( \tau \right) }{1-\exp \left[ \left( -\log (\mu )\right)^{\theta }\right]}}

Value

duchen gives the density, puchen gives the distribution function, quchen gives the quantile function and ruchen generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Korkmaz, M. C., Emrah, A., Chesneau, C. and Yousof, H. M., (2020). On the unit-Chen distribution with associated quantile regression and applications. Journal of Applied Statistics, 44(1) 1–22.

Examples

set.seed(123)
x <- ruchen(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Chen')
lines(S, duchen(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, puchen(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(quchen(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The unit-Half-Normal-E distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Half-Normal-E distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dughne(x, mu, theta, tau = 0.5, log = FALSE)

pughne(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qughne(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rughne(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\sqrt{\frac{2}{\pi }}\frac{\theta }{y\left[ -\log\left( y\right) \right] }\left( -{\frac{\log \left( y\right) }{\alpha }} \right)^{\theta }\mathrm{\exp }\left\{ -\frac{1}{2}\left[ -{\frac{\log \left( y\right) }{\alpha }}\right]^{2\theta }\right\}

Cumulative distribution function

F(y\mid \alpha ,\theta )=2\Phi \left[ -\left( -{\frac{\log \left( y\right) }{\alpha }}\right)^{\theta }\right]

Quantile function

Q(\tau \mid \alpha ,\theta )=\exp \left\{ -\alpha \left[ -\Phi^{-1}\left(\frac{\tau }{2}\right) \right]^{\frac{1}{\theta }}\right\}

Reparameterization

\alpha=g^{-1}(\mu )=-\log \left( \mu \right) \left[ -\Phi^{-1}\left( \frac{\tau }{2}\right) \right]^{-\frac{1}{\theta }}

Value

dughne gives the density, pughne gives the distribution function, qughne gives the quantile function and rughne generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Korkmaz, M. C., (2020). The unit generalized half normal distribution: A new bounded distribution with inference and application. University Politehnica of Bucharest Scientific, 82(2), 133–140.

Examples

set.seed(123)
x <- rughne(n = 1000, mu = 0.5, theta = 2, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Half-Normal-E')
lines(S, dughne(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pughne(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qughne(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

The unit-Half-Normal-X distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Half-Normal-X distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dughnx(x, mu, theta, tau = 0.5, log = FALSE)

pughnx(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qughnx(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rughnx(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\sqrt{\frac{2}{\pi }}\frac{\theta }{y\left(1-y\right) }\left( {\frac{y}{\alpha \left( 1-y\right) }}\right) ^{\theta }\mathrm{\exp }\left\{ -\frac{1}{2}\left[ {\frac{y}{\alpha \left( 1-y\right) }}\right] ^{2\theta }\right\}

Cumulative density function

F(y\mid \alpha ,\theta )=2\Phi \left[ \left( \frac{y}{\alpha \left(1-y\right) }\right) ^{\theta }\right] -1

Quantile Function

Q(\tau \mid \alpha )=\frac{\alpha \left[ \Phi ^{-1}\left( \frac{\tau +1}{2}\right) \right] ^{\frac{1}{\theta }}}{1+\alpha \left[ \Phi ^{-1}\left( \frac{ \tau +1}{2}\right) \right] ^{\frac{1}{\theta }}}

Reparametrization

\alpha=g^{-1}(\mu )=\frac{\mu }{\left( 1-\mu \right) \left[ \Phi ^{-1}\left( \frac{\tau +1}{2}\right) \right] ^{\frac{1}{\theta }}}

Value

dughnx gives the density, pughnx gives the distribution function, qughnx gives the quantile function and rughnx generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Bakouch, H. S., Nik, A. S., Asgharzadeh, A. and Salinas, H. S., (2021). A flexible probability model for proportion data: Unit-Half-Normal distribution. Communications in Statistics: CaseStudies, Data Analysis and Applications, 0(0), 1–18.

Examples

set.seed(123)
x <- rughnx(n = 1000, mu = 0.5, theta = 2, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Half-Normal-X')
lines(S, dughnx(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pughnx(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qughnx(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

The unit-Gompertz distribution

Description

Density function, distribution function, quantile function and random number deviates for the unit-Gompertz distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dugompertz(x, mu, theta, tau = 0.5, log = FALSE)

pugompertz(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qugompertz(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rugompertz(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\alpha \theta }{x}\exp \left\{ \alpha -\theta \log \left( y\right) -\alpha \exp \left[ -\theta \log \left( y\right) \right] \right\}

Cumulative density function

F(y\mid \alpha ,\theta )=\exp \left[ \alpha \left( 1-y^{\theta }\right) \right]

Quantile Function

Q(\tau \mid \alpha ,\theta )=\left[ \frac{\alpha -\log \left( \tau \right) }{\alpha }\right] ^{-\frac{1}{\theta }}

Reparameterization

\alpha =g^{-1}(\mu )=\frac{\log \left( \tau \right) }{1-\mu ^{\theta }}

Value

dugompertz gives the density, pugompertz gives the distribution function, qugompertz gives the quantile function and rugompertz generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Mazucheli, J., Menezes, A. F. and Dey, S., (2019). Unit-Gompertz Distribution with Applications. Statistica, 79(1), 25-43.

Examples

set.seed(123)
x <- rugompertz(n = 1000, mu = 0.5, theta = 2, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Gompertz')
lines(S, dugompertz(x = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pugompertz(q = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qugompertz(p = S, mu = 0.5, theta = 2, tau = 0.5), col = 2)

The unit-Gumbel distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Gumbel distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dugumbel(x, mu, theta, tau = 0.5, log = FALSE)

pugumbel(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qugumbel(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rugumbel(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\theta }{y(1-y)}\exp \left\{ -\alpha -\theta \log \left( \frac{y}{1-y}\right) -\exp \left[ -\alpha -\theta \log \left( \frac{y}{1-y}\right) \right] \right\}

Cumulative distribution function

F(y\mid\alpha,\theta)={\exp }\left[ -{{\exp }}\left( -\alpha \right)\left( \frac{1-y}{y}\right) ^{\theta } \right]

Quantile function

Q(\tau \mid \alpha, \theta)= \frac{\left [-\frac{1}{\log(\tau) }\right ]^{\frac{1}{\theta}}}{\exp\left ( \frac{\alpha}{\theta} \right )+\left [-\frac{1}{\log(\tau) }\right ]^{\frac{1}{\theta}}}

Reparameterization

\alpha = g^{-1}(\mu ) =\theta \log \left( {\frac{1-\mu }{\mu }}\right) +\log \left( -\frac{1}{\log \left( \tau \right) }\right)

where 0<y<1 and \theta >0 is the shape parameter.

Value

dugumbel gives the density, pugumbel gives the distribution function, qugumbel gives the quantile function and rugumbel generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli

Andre F. B. Menezes

References

Mazucheli, J. and Alves, B., (2021). The unit-Gumbel Quantile Regression Model for Proportion Data. Under Review.

Gumbel, E. J., (1941). The return period of flood flows. The Annals of Mathematical Statistics, 12(2), 163–190.

Examples

set.seed(6969)
x <- rugumbel(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.01)
hist(x, prob = TRUE, main = 'unit-Gumbel')
lines(S, dugumbel(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pugumbel(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qugumbel(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

The unit-Logistic distribution

Description

Density function, distribution function, quantile function and random number generation for the unit-Logistic distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

dulogistic(x, mu, theta, tau = 0.5, log = FALSE)

pulogistic(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

qulogistic(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

rulogistic(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\theta \exp \left( \alpha \right) \left(\frac{y}{1-y}\right) ^{\theta -1}}{\left[ 1+\exp \left( \alpha \right)\left( \frac{y}{1-y}\right) ^{\theta }\right] ^{2}}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\frac{\exp \left( \alpha \right) \left( \frac{y}{1-y}\right) ^{\theta }}{1+\exp \left( \alpha \right) \left( \frac{y}{1-y}\right) ^{\theta }}

Quantile function

Q(\tau \mid \alpha ,\theta )=\frac{\exp \left( -\frac{\alpha }{\theta }\right) \left( \frac{\tau }{1-\tau }\right) ^{\frac{1}{\theta }}}{1+\exp\left( -\frac{\alpha }{\theta }\right) \left( \frac{\tau }{1-\tau }\right) ^{ \frac{1}{\theta }}}

Reparameterization

\alpha=g^{-1}(\mu )=\log \left( \frac{\tau }{1-\tau }\right) -\theta \log \left( \frac{\mu }{1-\mu }\right)

Value

dulogistic gives the density, pulogistic gives the distribution function, qulogistic gives the quantile function and rulogistic generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli jmazucheli@gmail.com

André F. B. Menezes andrefelipemaringa@gmail.com

References

Paz, R. F., Balakrishnan, N. and Bazán, J. L., 2019. L-Logistic regression models: Prior sensitivity analysis, robustness to outliers and applications. Brazilian Journal of Probability and Statistics, 33(3), 455–479.

Examples

set.seed(123)
x <- rulogistic(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by =  0.01)
hist(x, prob = TRUE, main = 'unit-Logistic')
lines(S, dulogistic(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, pulogistic(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(qulogistic(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

Parametric unit quantile regression models

Description

Fit a collection of parametric unit quantile regression model by maximum likelihood using the log-likelihood function, the score vector and the hessian matrix implemented in C++.

Usage

unitquantreg(
  formula,
  data,
  subset,
  na.action,
  tau,
  family,
  link = c("logit", "probit", "cloglog", "cauchit"),
  link.theta = c("identity", "log", "sqrt"),
  start = NULL,
  control = unitquantreg.control(),
  model = TRUE,
  x = FALSE,
  y = TRUE
)

unitquantreg.fit(
  y,
  X,
  Z = NULL,
  tau,
  family,
  link,
  link.theta,
  start = NULL,
  control = unitquantreg.control()
)

Arguments

Details

The parameter estimation and inference are performed under the frequentist paradigm. The optimx R package is use, since allows different optimization technique to maximize the log-likelihood function. The analytical score function are use in the maximization and the standard errors are computed using the analytical hessian matrix, both are implemented in efficient away using C++.

Value

unitquantreg can return an object of class unitquantreg if tau is a scalar, i.e., a list with the following components.

While unitquantreg.fit returns an unclassed list with components up to elapsed_time.

If tau is a numeric vector with length greater than one an object of class unitquantregs is returned, which consist of list of objects of class unitquantreg for each specified quantiles.

Author(s)

André F. B. Menezes

Control parameters for unit quantile regression

Description

Auxiliary function that control fitting of unit quantile regression models using unitquantreg.

Usage

unitquantreg.control(
  method = "BFGS",
  hessian = FALSE,
  gradient = TRUE,
  maxit = 5000,
  factr = 1e+07,
  reltol = sqrt(.Machine$double.eps),
  trace = 0L,
  starttests = FALSE,
  dowarn = FALSE,
  ...
)

Arguments

Details

The control argument of unitquantreg uses the arguments of unitquantreg.control. In particular, the parameters in unitquantreg are estimated by maximum likelihood using the optimx, which is a general-purpose optimization wrapper function that calls other R tools for optimization, including the existing optim function. The main advantage of optimx is to unify the tools allowing a number of different optimization methods and provide sanity checks.

Value

A list with components named as the arguments.

Author(s)

André F. B. Menezes

References

Nash, J. C. and Varadhan, R. (2011). Unifying Optimization Algorithms to Aid Software System Users: optimx for R., Journal of Statistical Software, 43(9), 1–14.

See Also

optimx for more details about control parameters and unitquantreg.fit the fitting procedure used by unitquantreg.

Examples

data(sim_bounded, package = "unitquantreg")
sim_bounded_curr <- sim_bounded[sim_bounded$family == "uweibull", ]

# Fitting using the analytical gradient
fit_gradient <- unitquantreg(formula = y1 ~ x,
                             data = sim_bounded_curr, tau = 0.5,
                             family = "uweibull",
                             control = unitquantreg.control(gradient = TRUE,
                                                            trace = 1))

# Fitting without using the analytical gradient
fit_nogradient <- unitquantreg(formula = y1 ~ x,
                             data = sim_bounded_curr, tau = 0.5,
                             family = "uweibull",
                             control = unitquantreg.control(gradient = FALSE,
                                                            trace = 1))
# Compare estimated coefficients
cbind(gradient = coef(fit_gradient), no_gradient = coef(fit_nogradient))

The unit-Weibull distribution

Description

Density function, distribution function, quantile function and random number generation function for the unit-Weibull distribution reparametrized in terms of the \tau-th quantile, \tau \in (0, 1).

Usage

duweibull(x, mu, theta, tau = 0.5, log = FALSE)

puweibull(q, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

quweibull(p, mu, theta, tau = 0.5, lower.tail = TRUE, log.p = FALSE)

ruweibull(n, mu, theta, tau = 0.5)

Arguments

Details

Probability density function

f(y\mid \alpha ,\theta )=\frac{\alpha \theta }{y}\left[ -\log (y)\right]^{\theta -1}\exp \left\{ -\alpha \left[ -\log (y)\right]^{\theta }\right\}

Cumulative distribution function

F(y\mid \alpha ,\theta )=\exp \left\{ -\alpha \left[ -\log (y)\right]^{\theta }\right\}

Quantile function

Q\left( \tau \mid \alpha ,\theta \right) =\exp \left\{ -\left[ -\frac{\log (\tau )}{\alpha }\right]^{\frac{1}{\theta }}\right\}

Reparameterization

\alpha =g^{-1}(\mu )=-\frac{\log (\tau )}{[-\log (\mu )]^{\theta}}

Value

duweibull gives the density, puweibull gives the distribution function, quweibull gives the quantile function and ruweibull generates random deviates.

Invalid arguments will return an error message.

Author(s)

Josmar Mazucheli

André F. B. Menezes

References

Mazucheli, J., Menezes, A. F. B and Ghitany, M. E., (2018). The unit-Weibull distribution and associated inference. Journal of Applied Probability and Statistics, 13(2), 1–22.

Mazucheli, J., Menezes, A. F. B., Fernandes, L. B., Oliveira, R. P. and Ghitany, M. E., (2020). The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics, 47(6), 954–974.

Mazucheli, J., Menezes, A. F. B., Alqallaf, F. and Ghitany, M. E., (2021). Bias-Corrected Maximum Likelihood Estimators of the Parameters of the Unit-Weibull Distribution. Austrian Journal of Statistics, 50(3), 41–53.

Examples

set.seed(6969)
x <- ruweibull(n = 1000, mu = 0.5, theta = 1.5, tau = 0.5)
R <- range(x)
S <- seq(from = R[1], to = R[2], by = 0.01)
hist(x, prob = TRUE, main = 'unit-Weibull')
lines(S, duweibull(x = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(ecdf(x))
lines(S, puweibull(q = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)
plot(quantile(x, probs = S), type = "l")
lines(quweibull(p = S, mu = 0.5, theta = 1.5, tau = 0.5), col = 2)

Vuong test

Description

Performs Vuong test between two fitted objects of class unitquantreg

Usage

vuong.test(object1, object2, alternative = c("two.sided", "less", "greater"))

Arguments

Details

The statistic of Vuong likelihood ratio test for compare two non-nested regression models is defined by

T = \frac{1}{\widehat{\omega}^2\,\sqrt{n}}\,\sum_{i=1}^{n}\, \log\frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{ g(y_i \mid \boldsymbol{x}_i,\widehat{\boldsymbol{\gamma}})}

where

\widehat{\omega}^2 = \frac{1}{n}\,\sum_{i=1}^{n}\,\left(\log \frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}\right)^2 - \left[\frac{1}{n}\,\sum_{i=1}^{n}\,\left(\log \frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{ g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}\right)\right]^2

is an estimator for the variance of \frac{1}{\sqrt{n}}\,\displaystyle\sum_{i=1}^{n}\,\log\frac{f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}})}{g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}})}, f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}}) and g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}}) are the corresponding rival densities evaluated at the maximum likelihood estimates.

When n \rightarrow \infty we have that T \rightarrow N(0, 1) in distribution. Therefore, at \alpha\% level of significance the null hypothesis of the equivalence of the competing models is rejected if |T| > z_{\alpha/2}, where z_{\alpha/2} is the \alpha/2 quantile of standard normal distribution.

In practical terms, f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}}) is better (worse) than g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}}) if T>z_{\alpha/2} (or T< -z_{\alpha/2}).

Value

A list with class "htest" containing the following components:

Author(s)

André F. B. Menezes

Josmar Mazucheli

References

Vuong, Q. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57(2), 307–333.

Examples

data(sim_bounded, package = "unitquantreg")
sim_bounded_curr <- sim_bounded[sim_bounded$family == "uweibull", ]

fit_uweibull <- unitquantreg(formula = y1 ~ x, tau = 0.5,
                             data = sim_bounded_curr,
                             family = "uweibull")
fit_kum <- unitquantreg(formula = y1 ~ x, tau = 0.5,
                             data = sim_bounded_curr,
                             family = "kum")

ans <- vuong.test(object1 = fit_uweibull, object2 = fit_kum)
ans
str(ans)

Access to piped water supply data set

Description

The access of people in households with piped water supply in the cities of Brazil from the Southeast and Northeast regions. Information obtained during the census of 2010.

Usage

data(water, package = "unitquantreg")

Format

data.frame with 3457 observations and 5 columns:

• phpws: the proportion of households with piped water supply.

• mhdi: municipal human development index.

• incpc: per capita income.

• region: 0 for Southeast, 1 for Northeast.

• pop: total population.

Author(s)

André F. B. Menezes

References

Mazucheli, J., Menezes, A. F. B., Fernandes, L. B., Oliveira, R. P. and Ghitany, M. E., (2020). The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Jounal of Applied Statistics, 47(6), 954–974.