Package 'unitquantreg'

Title: Parametric Quantile Regression Models for Bounded Data
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>.
Authors: André F. B. Menezes [aut, cre] , Josmar Mazucheli [aut]
Maintainer: André F. B. Menezes <[email protected]>
License: Apache License (>= 2)
Version: 0.0.6
Built: 2025-03-05 04:49:42 UTC
Source: https://github.com/andrmenezes/unitquantreg

Help Index

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 [email protected]

Josmar Mazucheli [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

shape parameter.
tau

the parameter to specify which quantile use in the parametrization.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq x) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y;α,θ)=αθy1−y2arcsech(y)θ−1exp⁡[−αarcsech(y)θ]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;α,θ)=exp⁡[−αarcsech(y)θ]F(y;\alpha, \theta)=\exp\left [ -\alpha \mathrm{arcsech}(y)^\theta \right ]

Quantile function

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

Reparameterization

α=g−1(μ)=−log⁡(τ)arcsech(μ)θ\alpha = g^{-1}(\mu) = -\frac{\log(\tau)}{\mathrm{arcsech}(\mu)^\theta}

where θ>0\theta >0 is the shape parameter and arcsech(y)=log⁡[(1+1−y2)/y]\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

object

fitted model object of class unitquantreg.
...

currently not used.
nsim

number of simulations used to compute envelope. Default is 99.
halfnormal

logical. If TRUE, a half-normal plot is produced. If FALSE, a normal plot is produced.
plot

Should the (half-)normal plot be plotted? Default is TRUE.
output

Should the output be returned? Default is TRUE.
level

confidence level of the simulated envelope. Default is 0.95.
resid.type

type of residuals to be used. The default is quantile. See residuals.unitquantreg for further details.

Details

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

  1. generate sample set of nn 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 nn 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(i+n−0.1252n+0.5)G^{-1} \left(\frac{i + n - 0.125}{2n + 0.5} \right)

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

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

    for normal plots, i.e., halfnormal=FALSE, where G(⋅)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:

obs

the observed residuals.
teo

the theoretical residuals.
lower

lower envelope band.
median

median envelope band.
upper

upper envelope band.
time_elapsed

time elapsed to fit the nsim models.

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=θ2π1y(1−y)exp⁡{−12[α+θlog⁡(y1−y)]2}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∣α,θ)=Φ[α+θlog⁡(y1−y)]F(y\mid \alpha ,\theta )=\Phi \left[ \alpha +\theta \log \left( \frac{y}{1-y}\right) \right]

Quantile function

Q(τ∣α,θ)=exp⁡[Φ−1(τ)−αθ]1+exp⁡[Φ−1(τ)−αθ]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

α=g−1(μ)=Φ−1(τ)−θlog⁡(μ1−μ)\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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=αθyθ−1(1−yθ)α−1f(y\mid \alpha ,\theta )=\alpha \theta y^{\theta -1}(1-y^{\theta })^{\alpha-1}

Cumulative distribution function

F(y∣α,θ)=1−(1−yθ)αF(y\mid \alpha ,\theta )=1-\left( 1-y^{\theta }\right) ^{\alpha }

Quantile function

Q(τ∣α,θ)=[1−(1−τ)1α]1θQ(\tau \mid \alpha ,\theta )=\left[ 1-\left( 1-\tau \right) ^{\frac{1}{\alpha }}\right] ^{\frac{1}{\theta }}

Reparameterization

α=g−1(μ)=log⁡(1−τ)log⁡(1−μθ)\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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=θ(1+α)yθ−1(1+αyθ)2f(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∣α,θ)=(1+α)yθ1+αyθF(y\mid \alpha ,\theta )=\frac{\left( 1+\alpha \right) y^{\theta }}{1+\alpha y^{\theta }}

Quantile function

Q(τ∣α,θ)=[τ1+α(1−τ)]1θQ(\tau \mid \alpha ,\theta )=\left[ \frac{\tau }{1+\alpha \left( 1-\tau\right) }\right] ^{\frac{1}{\theta }}

Reparameterization

α=g−1(μ)=−1−τμθ(1−τ)\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 [email protected]

André F. B. Menezes [email protected]

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

...

unitquantreg objects separated by commas. Not use in print method.
lt

a list with one or more unitquantreg objects.
x

object of class likelihood_stats obtained from likelihood_stats function.

Details

Neg2LogLike: The log-likelihood is reported as

Neg2LogLike=−2log⁡(L)Neg2LogLike= -2\log(L)

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

AIC=−2log⁡(L)+2pAIC = -2\log(L)+2p

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

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

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

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

where LL is the likelihood function.

Value

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

call

the matched call.
stats

ordered matrix according AIC value containg the likelihood based statistics.

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

par

vector of regression model coefficients for μ\mu and/or θ\theta.
tau

quantile level, value between 0 and 1.
family

specify the distribution family name.
linkobj, linkobj.theta

a function, usually obtained from make.link for link function of μ\mu and θ\theta, respectively.
X

design matrix related to the μ\mu parameter.
Z

design matrix related to the θ\theta parameter.
y

vector of response variable.

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

digits

minimal number of significant digits.
...

additional argument(s) for methods. Currently not used.
object, x

fitted model object of class unitquantreg.
correlation

logical; if TRUE, the correlation matrix of the estimated parameters is returned and printed. Default is FALSE.
type

character indicating type of fitted values to return.
parm

a specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.
level

the confidence level required.
formula

an R formula.
formula.

Changes to the formula see update.formula for details.
evaluate

If true evaluate the new call else return the call.

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

...

unitquantreg objects separated by commas.
lt

a list with one or more unitquantreg objects.
p.adjust.method

a character string specifying the method for multiple testing adjustment; almost always one of p.adjust.methods. Can be abbreviated.
alternative

indicates the alternative hypothesis and must be one of "two.sided" (default), "less", or "greater". Can be abbreviated.

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

x

fitted model object of class unitquantreg.
which

integer. if a subset of the plots is required, specify a subset of the numbers 1 to 4, see below for further details.
caption

character. Captions to appear above the plots.
sub.caption

character. Common title-above figures if there are multiple.
main

character. Title to each plot in addition to the above caption.
ask

logical. If TRUE, the user is asked before each plot.
...

other parameters to be passed through to plotting functions.
add.smooth

logical. Indicates if a smoother should be added to most plots
type

character. Indicates type of residual to be used, see residuals.unitquantreg.
nsim

integer. Number of simulations in half-normal plots, see hnp.unitquantreg.
level

numeric. Confidence level of the simulated envelope, see hnp.unitquantreg.

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

x

fitted model object of class unitquantregs.
which

character. Indicate the type of plot. Currently supported are "coef" which provide the estimated coefficients for several quantiles and "conddist" which provide the conditional distribution (cdf or pdf) at specific values of covariates.
output_df

logical. Should data.frame used to plot be returned?
parm

a specification of which parameters are to be plotted, either a vector of numbers or a vector of names. By default, all parameters are considered.
level

level of significance for the confidence interval of parameters.
mean_effect

logical. Should a line for the mean effect coefficients be added?
mfrow, mar, ylim, main, col, border, cex, pch, type, xlab, ylab

graphical parameters.
dist_type

character. Which conditional distribution should be plotted? The options are "density" or "cdf".
at_avg

logical. Should consider the conditional distribution at average values of covariates?
at_obs

list. List with name and values for each covariate.
legend_position

character. The legend position argument used in legend function.
...

other parameters to be passed through to plotting functions.

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

object

fitted model object of class unitquantreg.
newdata

optionally, a data frame in which to look for variables with which to predict. If omitted, the original observations are used.
type

character indicating type of predictions. The options are link, quantile, shape and terms.
interval

type of interval desired. The options are none and confidence. The "terms" option returns a matrix giving the fitted values of each term in the model formula on the linear predictor scale.
level

coverage probability for the confidence intervals. Default is 0.95.
se.fit

logical. If TRUE return the asymptotic standard errors.
...

currently not used.

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:

fit

Predictions, as for se.fit = FALSE.
se.fit

Estimated standard errors.

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

object

fitted model object of class unitquantreg.
type

character indicating type of residuals. The options are "quantile", "cox-snell", "working" and "partial".
...

currently not used.

Details

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

rQ=Φ−1[F(yi∣μ^i,θ^i)]r_{Q} = \Phi^{-1}\left[ F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]

and

rCS=−log⁡[1−F(yi∣μ^i,θ^i)]r_{CS} = -\log\left[1- F(y_i \mid \widehat{\mu}_i, \widehat{\theta}_i)\right]

where μ^i\widehat{\mu}_i and θ^i\widehat{\theta}_i are the fitted values of parameters μ\mu and θ\theta, F(⋅∣⋅,⋅)F(\cdot \mid \cdot, \cdot) is the cumulative distribution function (c.d.f.) and Φ(⋅)\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, rQr_Q, follow a standard Normal distribution and the Cox-Snell residuals, rCSr_{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., τ=0.5\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, θ=2\theta = 2.

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

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

  • z: covariate related to θi\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 μ\mu and one covariate simulated from a standard uniform distribution, where the true vector of coefficients for μ\mu is β=(1,2)\boldsymbol{\beta} = (1, 2) and θ=2\theta = 2. The latter was simulated assuming a regression structure for both μ\mu and θ\theta (shape parameter) and only one independent covariates simulated from two standard uniform distributions. The true vectors of coefficients for μ\mu and θ\theta are β=(1,2)\boldsymbol{\beta} = (1, 2) and γ=(−1,1)\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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=12yαθ2π[(−αlog⁡(y))12+(−αlog⁡(y))32]exp⁡[12θ2(2+log⁡(y)α+αlog⁡(y))]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∣α,θ)=1−Φ{1θ[(−log⁡(y)α)12−(−αlog⁡(y))12]}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(τ∣α,θ)=exp⁡{−2α2+[θΦ−1(1−τ)]2−θΦ−1(1−τ)4+[θΦ−1(1−τ)]2}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

α=g−1(μ)=log⁡(μ)g(θ,τ)\alpha=g^{-1}(\mu )=\log \left( \mu \right) g\left( \theta ,\tau \right)

where g(θ,τ)=−12{2+[θΦ−1(1−τ)]2−θΦ−1(1−τ)4+θΦ−1(1−τ)}.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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=αθy[−log⁡(y)]θ−1{1+[−log⁡(y)]θ}−α−1f(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∣α,θ)={1+[−log⁡(y)]θ}−αF(y\mid \alpha, \theta )=\left\{ 1+\left[ -\log (y)\right] ^{\theta}\right\} ^{-\alpha }

Quantile function

Q(τ∣α,θ)=exp⁡[−(τ−1α−1)1θ]Q(\tau \mid \alpha, \theta )=\exp \left[ -\left( \tau ^{-\frac{1}{\alpha }}-1\right)^{\frac{1}{\theta }} \right]

Reparameterization

α=g−1(μ)=log⁡(τ−1)log⁡[1+log⁡(1μ)θ]\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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=αθy[−log⁡(y)]θ−1exp⁡{[−log⁡(y)]θ}exp⁡{α{1−exp⁡[(−log⁡(y))θ]}}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∣α,θ)=exp⁡{α{1−exp⁡[(−log⁡(y))θ]}}F(y\mid \alpha ,\theta )=\exp \left\{ \alpha \left\{ 1-\exp \left[ \left(-\log (y)\right)^{\theta }\right] \right\} \right\}

Quantile function

Q(τ∣α,θ)=exp⁡{−[log⁡(1−log⁡(τ)α)]1θ}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

α=g−1(μ)=log⁡(τ)1−exp⁡[(−log⁡(μ))θ]\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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=2πθy[−log⁡(y)](−log⁡(y)α)θexp⁡{−12[−log⁡(y)α]2θ}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∣α,θ)=2Φ[−(−log⁡(y)α)θ]F(y\mid \alpha ,\theta )=2\Phi \left[ -\left( -{\frac{\log \left( y\right) }{\alpha }}\right)^{\theta }\right]

Quantile function

Q(τ∣α,θ)=exp⁡{−α[−Φ−1(τ2)]1θ}Q(\tau \mid \alpha ,\theta )=\exp \left\{ -\alpha \left[ -\Phi^{-1}\left(\frac{\tau }{2}\right) \right]^{\frac{1}{\theta }}\right\}

Reparameterization

α=g−1(μ)=−log⁡(μ)[−Φ−1(τ2)]−1θ\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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=2πθy(1−y)(yα(1−y))θexp⁡{−12[yα(1−y)]2θ}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∣α,θ)=2Φ[(yα(1−y))θ]−1F(y\mid \alpha ,\theta )=2\Phi \left[ \left( \frac{y}{\alpha \left(1-y\right) }\right) ^{\theta }\right] -1

Quantile Function

Q(τ∣α)=α[Φ−1(τ+12)]1θ1+α[Φ−1(τ+12)]1θQ(\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

α=g−1(μ)=μ(1−μ)[Φ−1(τ+12)]1θ\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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to be used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=αθxexp⁡{α−θlog⁡(y)−αexp⁡[−θlog⁡(y)]}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∣α,θ)=exp⁡[α(1−yθ)]F(y\mid \alpha ,\theta )=\exp \left[ \alpha \left( 1-y^{\theta }\right) \right]

Quantile Function

Q(τ∣α,θ)=[α−log⁡(τ)α]−1θQ(\tau \mid \alpha ,\theta )=\left[ \frac{\alpha -\log \left( \tau \right) }{\alpha }\right] ^{-\frac{1}{\theta }}

Reparameterization

α=g−1(μ)=log⁡(τ)1−μθ\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 [email protected]

André F. B. Menezes [email protected]

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile use in the parametrization.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq x) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=θy(1−y)exp⁡{−α−θlog⁡(y1−y)−exp⁡[−α−θlog⁡(y1−y)]}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∣α,θ)=exp⁡[−exp⁡(−α)(1−yy)θ]F(y\mid\alpha,\theta)={\exp }\left[ -{{\exp }}\left( -\alpha \right)\left( \frac{1-y}{y}\right) ^{\theta } \right]

Quantile function

Q(τ∣α,θ)=[−1log⁡(τ)]1θexp⁡(αθ)+[−1log⁡(τ)]1θQ(\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

α=g−1(μ)=θlog⁡(1−μμ)+log⁡(−1log⁡(τ))\alpha = g^{-1}(\mu ) =\theta \log \left( {\frac{1-\mu }{\mu }}\right) +\log \left( -\frac{1}{\log \left( \tau \right) }\right)

where 0<y<10<y<1 and θ>0\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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile is to used.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq{x}) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=θexp⁡(α)(y1−y)θ−1[1+exp⁡(α)(y1−y)θ]2f(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∣α,θ)=exp⁡(α)(y1−y)θ1+exp⁡(α)(y1−y)θF(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(τ∣α,θ)=exp⁡(−αθ)(τ1−τ)1θ1+exp⁡(−αθ)(τ1−τ)1θQ(\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

α=g−1(μ)=log⁡(τ1−τ)−θlog⁡(μ1−μ)\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 [email protected]

André F. B. Menezes [email protected]

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

formula

symbolic description of the quantile model like y ~ x or y ~ x | z. See below for details.
data

data.frame contain the variables in the model.
subset

an optional vector specifying a subset of observations to be used in the fitting process.
na.action

a function which indicates what should happen when the data contain NAs.
tau

numeric vector. The quantile(s) to be estimated, i.e., number between 0 and 1. If just one quantile is specified an object of class unitquantreg is returned. If a numeric vector of values between 0 and 1 is specified an object of class unitquantregs is returned. See below for details.
family

character. Specify the distribution family.
link

character. Specify the link function in the quantile model. Currently supported are logit, probit, cloglog and cauchit. Default is logit.
link.theta

character. Specify the link function in the shape model. Currently supported are identity, log and sqrt. Default is log.
start

numeric vector. An optional vector with starting values for all parameters.
control

list. Control arguments specified via unitquantreg.control.
model

logical. Indicates whether model frame should be included as a component of the returned value.
x, y

logical. If TRUE the corresponding components of the fit (model frame, response, model matrix) are returned. For unitquantreg.fit y should be the numeric response vector with values in (0,1).
X, Z

numeric matrix. Regressor matrix for the quantile and shape model, respectively. Default is constant shape model, i.e., Z is matrix with column of ones.

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.

family

the distribution family name.
coefficients

a list with elements "quantile" and "shape" containing the coefficients from the respective models.
fitted.values

a list with elements "quantile" and "shape" containing the fitted parameters from the respective models.
linear.predictors

a list with elements "quantile" and "shape" containing the fitted linear predictors from the respective models.
link

a list with elements "quantile" and "shape" containing the link objects from the respective models.
tau

the quantile specify.
loglik

log-likelihood of the fitted model.
gradient

gradient evaluate at maximum likelihood estimates.
vcov

covariance matrix of all parameters in the model.
nobs

number of observations.
npar

number of parameters.
df.residual

residual degrees of freedom in the fitted model.
theta_const

logical indicating if the θ\theta parameter was treated as nuisance parameter.
control

the control parameters used to fit the model.
iterations

number of iterations of optimization method.
converged

logical, if TRUE indicates successful convergence.
kkt

a list of logical kkt1 and kkt2 provide check on Kuhn-Karush-Tucker conditions, first-order KKT test (kkt1) checks whether the gradient at the final parameters estimates is "small" and the second-order KKT test (kkte) checks whether the Hessian at the final parameters estimates is positive definite.
elapsed_time

time elapsed to fit the model.
call

the original function call.
formula

the original model formula.
terms

a list with elements "quantile", "shape" and "full" containing the terms objects for the respective models.
model

the full model frame, if model = TRUE.
y

the response vector, if y = TRUE.
x

a list with elements "quantile" and "shape" containing the model matrices from the respective models, if x = TRUE.

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

method

string. Specify the method argument passed to optimx.
hessian

logical. Should use the numerically Hessian matrix to compute variance-covariance? Default is FALSE, i.e., use the analytic Hessian.
gradient

logical. Should use the analytic gradient? Default is TRUE.
maxit

integer. Specify the maximal number of iterations passed to optimx.
factr

numeric.Controls the convergence of the "L-BFGS-B" method.
reltol

numeric. Relative convergence tolerance passed to optimx.
trace

non-negative integer. If positive, tracing information on the progress of the optimization is produced.
starttests

logical. Should optimx run tests of the functions and parameters? Default is FALSE.
dowarn

logical. Show warnings generated by optimx? Default is FALSE.
...

arguments passed to optimx.

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 τ\tau-th quantile, τ∈(0,1)\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

x, q

vector of positive quantiles.
mu

location parameter indicating the τ\tau-th quantile, τ∈(0,1)\tau \in (0, 1).
theta

nonnegative shape parameter.
tau

the parameter to specify which quantile use in the parametrization.
log, log.p

logical; If TRUE, probabilities p are given as log(p).
lower.tail

logical; If TRUE, (default), P(X≤x)P(X \leq x) are returned, otherwise P(X>x)P(X > x).
p

vector of probabilities.
n

number of observations. If length(n) > 1, the length is taken to be the number required.

Details

Probability density function

f(y∣α,θ)=αθy[−log⁡(y)]θ−1exp⁡{−α[−log⁡(y)]θ}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∣α,θ)=exp⁡{−α[−log⁡(y)]θ}F(y\mid \alpha ,\theta )=\exp \left\{ -\alpha \left[ -\log (y)\right]^{\theta }\right\}

Quantile function

Q(τ∣α,θ)=exp⁡{−[−log⁡(τ)α]1θ}Q\left( \tau \mid \alpha ,\theta \right) =\exp \left\{ -\left[ -\frac{\log (\tau )}{\alpha }\right]^{\frac{1}{\theta }}\right\}

Reparameterization

α=g−1(μ)=−log⁡(τ)[−log⁡(μ)]θ\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

object1, object2

objects of class unitquantreg containing the fitted models.
alternative

indicates the alternative hypothesis and must be one of "two.sided" (default), "less", or "greater". You can specify just the initial letter of the value, but the argument name must be given in full. See ‘Details’ for the meanings of the possible values.

Details

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

T=1ω^2 n ∑i=1n log⁡f(yi∣xi,θ^)g(yi∣xi,γ^)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

ω^2=1n ∑i=1n (log⁡f(yi∣xi,θ^)g(yi∣xi,γ^))2−[1n ∑i=1n (log⁡f(yi∣xi,θ^)g(yi∣xi,γ^))]2\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 1n ∑i=1n log⁡f(yi∣xi,θ^)g(yi∣xi,γ^)\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(yi∣xi,θ^)f(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\theta}}) and g(yi∣xi,γ^)g(y_i \mid \boldsymbol{x}_i, \widehat{\boldsymbol{\gamma}}) are the corresponding rival densities evaluated at the maximum likelihood estimates.

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

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

Value

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

statistic

the value of the test statistic.
p.value

the p-value of the test.
alternative

a character string describing the alternative hypothesis.
method

a character string with the method used.
data.name

a character string ginven the name of families models under comparison.

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.