Package {qrcmNP}

Type:	Package
Title:	Nonlinear and Penalized Quantile Regression Coefficients Modeling
Version:	1.0.0
Description:	Penalized and nonlinear parametric modeling of quantile regression coefficient functions. The penalized integrated loss minimization estimator (lasso and group lasso penalties) of Sottile, Frumento, Chiodi and Bottai (2020) <doi:10.1177/1471082X19825523> performs covariate selection using the information on all quantiles simultaneously. Nonlinear parametric quantile models, in which the conditional quantile function can be a nonlinear function of the parameters, of the covariates or both, are estimated by integrated loss minimization as in Bottai and Cilluffo (2020) <doi:10.1177/0962280220941159>. Both approaches extend the quantile regression coefficients modeling framework of Frumento and Bottai (2016) <doi:10.1111/biom.12410> implemented in 'qrcm'.
License:	GPL-2
URL:	https://github.com/gianluca-sottile/qrcmNP
BugReports:	https://github.com/gianluca-sottile/qrcmNP/issues
Depends:	R (≥ 4.1.0)
Imports:	graphics, grDevices, qrcm (≥ 3.0), stats, survival (≥ 2.41), utils
Suggests:	knitr, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder:	knitr
Config/testthat/edition:	3
Encoding:	UTF-8
Language:	en-US
Config/roxygen2/version:	8.1.1
NeedsCompilation:	no
Packaged:	2026-10-29 13:49:45 UTC; gianlucasottile
Author:	Gianluca Sottile [aut, cre], Paolo Frumento [aut, cph] (author of parts of the code adapted from 'qrcm')
Maintainer:	Gianluca Sottile <gianluca.sottile@unipa.it>
Repository:	CRAN
Date/Publication:	2026-10-30 09:50:07 UTC

qrcmNP: Nonlinear and Penalized Quantile Regression Coefficients Modeling

Description

Quantile regression coefficients modeling (QRCM, Frumento and Bottai, 2016) describes the regression coefficients \beta(p) of a quantile regression model as parametric functions of the quantile order p, and estimates all quantiles simultaneously by minimizing an integrated loss function. 'qrcmNP' extends this framework in two directions:

Details

• piqr(): penalized QRCM for covariate selection (Sottile et al., 2020), with lasso and group lasso penalties, tuning by AIC/BIC (gof.piqr()) or cross-validation (cv_piqr()), and post-selection inference (refit_piqr());

• niqr(): nonlinear parametric quantile models, in which the quantile function can be a nonlinear function of the parameters, of the covariates, or both (Bottai and Cilluffo, 2020), with sandwich, cluster-robust and bootstrap standard errors, information criteria (gof.niqr()) and a goodness-of-fit test (test.fit.niqr()).

Author(s)

Maintainer: Gianluca Sottile gianluca.sottile@unipa.it (ORCID)

Authors:

• Gianluca Sottile gianluca.sottile@unipa.it (ORCID)

• Paolo Frumento (author of parts of the code adapted from 'qrcm') [copyright holder]

References

Frumento, P. and Bottai, M. (2016). Parametric modeling of quantile regression coefficient functions. Biometrics, 72(1), 74-84. doi:10.1111/biom.12410

Sottile, G., Frumento, P., Chiodi, M. and Bottai, M. (2020). A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20(4), 369-385. doi:10.1177/1471082X19825523

Bottai, M. and Cilluffo, G. (2020). Nonlinear parametric quantile models. Statistical Methods in Medical Research, 29(12), 3757-3769. doi:10.1177/0962280220941159

See Also

Useful links:

• https://github.com/gianluca-sottile/qrcmNP

• Report bugs at https://github.com/gianluca-sottile/qrcmNP/issues

Extract Coefficients of a Penalized QRCM

Description

Extract Coefficients of a Penalized QRCM

Usage

## S3 method for class 'piqr'
coef(object, pos.lambda, lambda, ...)

Arguments

Value

The q \times k matrix \hat\theta at the selected \lambda.

See Also

piqr()

Cross-Validation for Penalized QRCM

Description

K-fold cross-validation of the integrated loss along the path of a piqr() fit (uncensored data only).

Usage

cv_piqr(
  formula,
  formula.p = ~slp(p, 3),
  weights,
  data,
  s,
  penalty = c("lasso", "group"),
  penalty.factor,
  nlambda = 100,
  lambda.min.ratio,
  lambda,
  nfolds = 10,
  foldid,
  ...
)

Arguments

Details

The cross-validated loss of observation i is \int_0^1 \rho_p\{y_i - x_i^\top\hat\theta_{-i}\, b(p)\}\,dp, computed numerically, where \hat\theta_{-i} is estimated without the fold containing i. In each fold the variables are standardized as in the full data and \lambda is rescaled by the size of the training sample, so that the same value of \lambda corresponds to the same penalty in all fits. lambda.1se is the largest \lambda whose error is within one standard error of the minimum.

Value

An object of class "cv.piqr", a list with items lambda, cvm (mean cross-validated loss), cvsd (its standard error), lambda.min, lambda.1se, pos.min, pos.1se (positions in lambda), foldid and fit (the piqr() fit on the full data).

See Also

piqr(), gof.piqr()

Examples

set.seed(1)
n <- 300
dat <- data.frame(x1 = runif(n), x2 = runif(n), x3 = runif(n))
dat$y <- 1 + qnorm(runif(n)) + 2 * dat$x1
cv <- cv_piqr(y ~ x1 + x2 + x3, formula.p = ~ slp(p, 2), data = dat,
              nfolds = 5, nlambda = 20)
cv
plot(cv)
coef(cv$fit, pos.lambda = cv$pos.1se)

Information Criteria for Nonlinear Parametric Quantile Models

Description

Computes the goodness-of-fit criteria of Bottai and Cilluffo (2020, Section 3) for one or more niqr() fits.

Usage

gof.niqr(object, ...)

Arguments

Details

With L_n the minimized (average) loss, k the number of parameters and n the sample size, the criteria are AIC^* = \log L_n + k/n and BIC^* = \log L_n + k\log(n)/(2n) (AIC.log, BIC.log), and AIC = nL_n + 2k, BIC = nL_n + k\log n. Smaller values indicate a better fit. Bottai and Cilluffo (2020) write the last two criteria in terms of L_n; here the total loss nL_n is used, otherwise the penalty would dominate. Note that, unlike the log versions, AIC and BIC depend on the scale of the response; the log versions are recommended.

Value

A data frame with columns npar, loss, AIC, BIC, AIC.log and BIC.log (one row per model).

References

Bottai, M. and Cilluffo, G. (2020). Nonlinear parametric quantile models. Statistical Methods in Medical Research, 29(12), 3757-3769. doi:10.1177/0962280220941159

See Also

niqr(), test.fit.niqr()

Select the Tuning Parameter of a Penalized QRCM by AIC or BIC

Description

Computes an information criterion along the path of a piqr() fit and returns the value of \lambda at which it is minimized.

Usage

gof.piqr(
  object,
  method = c("BIC", "AIC"),
  Cn = 1,
  plot = TRUE,
  logi = TRUE,
  df = NULL,
  ...
)

Arguments

Details

With logi = TRUE, the criteria of Sottile et al. (2020, eq. 3.6-3.7) are

AIC(\lambda) = \log L(\hat\theta_\lambda) + 2\,df(\lambda)/n,\qquad BIC(\lambda) = \log L(\hat\theta_\lambda) + C_n\log(n)\,df(\lambda)/n,

where L is the minimized integrated loss. With logi = FALSE, AIC = L + 2\,df and BIC = L + C_n\log(n)\,df. Note that Bottai and Cilluffo (2020) define the log-criteria with df/n and \log(n)df/(2n); the selected model is the same as with the definition above when C_n = 1/2 for the BIC.

Value

A list with items call, minLambda (the selected value of \lambda), dfMinLambda (its degrees of freedom), posMinLambda (its position in object$lambda), betaMin (the coefficients at the selected \lambda) and curve (the criterion along the path).

References

Sottile, G., Frumento, P., Chiodi, M. and Bottai, M. (2020). A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20(4), 369-385. doi:10.1177/1471082X19825523

Lee, E. R., Noh, H. and Park, B. U. (2014). Model selection via Bayesian information criterion for quantile regression models. Journal of the American Statistical Association, 109(505), 216-229. doi:10.1080/01621459.2013.836975

See Also

piqr(), cv_piqr()

Examples

set.seed(1)
n <- 200
x1 <- runif(n); x2 <- runif(n)
y <- 1 + qnorm(runif(n)) + 2 * x1
fit <- piqr(y ~ x1 + x2, formula.p = ~ slp(p, 2), nlambda = 20)
gof.piqr(fit, method = "AIC")
gof.piqr(fit, method = "BIC", df = "parameters", plot = FALSE)$posMinLambda

Nonlinear Parametric Quantile Models

Description

Estimates a parametric model for the whole conditional quantile function, possibly nonlinear in the parameters, in the covariates, or both, by minimizing the integrated loss function of Frumento and Bottai (2016), as described by Bottai and Cilluffo (2020).

Usage

niqr(
  formula,
  data,
  fun,
  fun2 = NULL,
  x0,
  weights,
  cluster,
  control = niqr.control(),
  ...
)

Arguments

Details

Let Q(p \mid x, \theta) be a parametric model for the conditional quantile function of y given x. The estimator \hat\theta minimizes

L_n(\theta) = \frac{1}{n}\sum_{i=1}^n w_i \int_0^1 \rho_p\{y_i - Q(p \mid x_i, \theta)\}\,dp,

where \rho_p(u) = u\{p - I(u < 0)\}. The integral is computed with the trapezoidal rule on a grid of n.p values of p (denser in the tails), splitting the interval containing p^*_i = F(y_i \mid x_i, \theta), where the integrand is not differentiable. The gradient -n^{-1}\sum_i w_i \int_0^1 \{p - I(p \ge p^*_i)\}\,\partial_\theta Q(p \mid x_i, \theta)\,dp is computed in the same way, with numerical derivatives of fun (or of fun2) with respect to \theta. The loss is minimized by a damped Newton algorithm that uses the Hessian

H_n(\theta) = \frac1n\sum_i w_i\,\frac{\partial_\theta Q(p^*_i \mid x_i, \theta)\, \partial_\theta Q(p^*_i \mid x_i, \theta)^\top}{Q'(p^*_i \mid x_i, \theta)},

where Q' is the derivative with respect to p, i.e. the inverse of the conditional density. H_n is positive semi-definite and estimates the Hessian of the expected loss at the true parameter.

The asymptotic covariance matrix of \hat\theta is estimated by the sandwich formula H^{-1}\Omega H^{-1}/n (Bottai and Cilluffo, 2020, Theorem 3), where \Omega is the covariance of the individual contributions to the gradient (summed within clusters if cluster is supplied), or by the (cluster) bootstrap (niqr.control(se = "boot")).

The model is assumed to be monotonically increasing in p for all observations. A warning is given if quantile crossing is detected. Good starting values x0 are important: the loss function may have several local minima when Q is not one-to-one in \theta (Bottai and Cilluffo, 2020, Theorem 1).

Value

An object of class "niqr", a list with items:

Note

The matrix interface niqr(fun, fun2, x0, X, y) of 'qrcmNP' 0.2.x was removed in version 1.0.0: use niqr(y ~ x, data, fun, fun2, x0), where the model matrix of the formula plays the role of the old X.

References

Bottai, M. and Cilluffo, G. (2020). Nonlinear parametric quantile models. Statistical Methods in Medical Research, 29(12), 3757-3769. doi:10.1177/0962280220941159

Frumento, P. and Bottai, M. (2016). Parametric modeling of quantile regression coefficient functions. Biometrics, 72(1), 74-84. doi:10.1111/biom.12410

See Also

niqr.control(), summary.niqr(), predict.niqr(), plot.niqr(), gof.niqr(), test.fit.niqr().

Examples

## Model linear in the covariates, nonlinear in the parameters:
## beta0(p) = theta1 + exp(theta2 * p), beta1(p) = theta3 + theta4 * p
set.seed(1234)
n <- 300
x <- runif(n)
fun <- function(theta, p) {
  cbind(theta[1] + exp(theta[2] * p), theta[3] + theta[4] * p)
}
u <- runif(n)
beta <- fun(c(1, 1, 1, 1), u)
y <- beta[, 1] + beta[, 2] * x
m <- niqr(y ~ x, fun = fun, x0 = c(a = 0.5, b = 0.5, c = 0.5, d = 0.5))
summary(m)
predict(m, type = "QF", p = c(0.1, 0.5, 0.9), newdata = data.frame(x = 0.5))

## Model nonlinear in the covariate: Hill dose-response quantile function
## Q(p | dose) = b0(p) + (b1 - b0(p)) / (1 + 10^(b3 - log10(dose)))

set.seed(1)
dose <- rep(10^seq(-2, 2, length.out = 20), each = 15)
fun.h <- function(theta, p) {
  cbind(theta[1] + exp(theta[2]) * qnorm(p), theta[3], theta[4])
}
fun2.h <- function(beta, X) {
  beta[, 1] + (beta[, 2] - beta[, 1]) / (1 + 10^(beta[, 3] - log10(X[, 2])))
}
yh <- fun2.h(fun.h(c(1, log(0.5), 10, 0), runif(length(dose))), cbind(1, dose))
mh <- niqr(yh ~ dose, fun = fun.h, fun2 = fun2.h, x0 = c(0, 0, 5, 0.5))
summary(mh)

Control Parameters for niqr

Description

Control Parameters for niqr

Usage

niqr.control(
  tol = 1e-06,
  maxit = 100,
  n.p = 199,
  eps = 1e-04,
  h = .Machine$double.eps^(1/3),
  se = c("sandwich", "boot", "none"),
  R = 200,
  verbose = FALSE
)

Arguments

Value

A list of control parameters.

See Also

niqr()

Penalized Quantile Regression Coefficients Modeling

Description

Fits the penalized integrated loss minimization (PILM) estimator of Sottile et al. (2020) along a path of values of the tuning parameter \lambda. Quantile regression coefficients are modelled as parametric functions of the quantile order p, and covariates are selected using the information on all quantiles simultaneously.

Usage

piqr(
  formula,
  formula.p = ~slp(p, 3),
  weights,
  data,
  s,
  nlambda = 100,
  lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
  lambda,
  tol = 1e-06,
  maxit = 100,
  penalty = c("lasso", "group"),
  penalty.factor,
  method = c("pnewton", "legacy"),
  verbose = FALSE,
  ...
)

Arguments

Details

Under the quantile regression coefficients model (QRCM) of Frumento and Bottai (2016), the conditional quantile function is

Q(p \mid x, \theta) = x^\top \theta\, b(p),

where b(p) is a set of k known functions of p (defined by formula.p) and \theta is a q \times k matrix of parameters. The penalized estimator minimizes

L_{PEN}(\theta) = \int_0^1 \sum_i w_i \rho_p\{y_i - Q(p \mid x_i, \theta)\}\,dp + \lambda\,\mathrm{pen}(\theta),

where \rho_p is the check loss of quantile regression. As suggested by Sottile et al. (2020), x and b(p) are standardized, and the response is mapped to the interval [0, 10]; the values of lambda refer to this internal scale. The estimated coefficients are returned on the original scale.

With method = "pnewton", for each \lambda the integrated loss is replaced by its second-order expansion (gradient and Hessian are available in closed form) and the resulting penalized quadratic problem is solved by coordinate descent ("lasso") or block coordinate descent ("group"); a backtracking line search guarantees the decrease of the penalized objective (proximal Newton method). The path starts from \lambda_{max}, the smallest value at which all penalized coefficients are zero, and uses warm starts, sequential strong rules and a final check of the Karush-Kuhn-Tucker conditions. With censored or truncated data, the estimating equations of Frumento and Bottai (2017) replace the gradient of the loss; since they are not the gradient of a loss function, the symmetric part of their Jacobian is used in the quadratic approximation, and the iterate with the smallest violation of the KKT conditions is returned (convergence is declared when the average violation is below 5e-4).

The intercept (the first row of \theta) is never penalized. Use gof.piqr() to select \lambda by AIC or BIC, cv_piqr() for cross-validation, and refit_piqr() to re-estimate the selected model without penalty and obtain standard errors (Sottile et al., 2020, Section 3.2).

Value

An object of class "piqr", a list with items:

References

Sottile, G., Frumento, P., Chiodi, M. and Bottai, M. (2020). A penalized approach to covariate selection through quantile regression coefficient models. Statistical Modelling, 20(4), 369-385. doi:10.1177/1471082X19825523

Frumento, P. and Bottai, M. (2016). Parametric modeling of quantile regression coefficient functions. Biometrics, 72(1), 74-84. doi:10.1111/biom.12410

Frumento, P. and Bottai, M. (2017). Parametric modeling of quantile regression coefficient functions with censored and truncated data. Biometrics, 73(4), 1179-1188. doi:10.1111/biom.12675

Friedman, J., Hastie, T. and Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent. Journal of Statistical Software, 33(1), 1-22. doi:10.18637/jss.v033.i01

Lee, J. D., Sun, Y. and Saunders, M. A. (2014). Proximal Newton-type methods for minimizing composite functions. SIAM Journal on Optimization, 24(3), 1420-1443. doi:10.1137/130921428

See Also

gof.piqr(), cv_piqr(), refit_piqr(), summary.piqr(), plot.piqr(), predict.piqr().

Examples

set.seed(1234)
n <- 300
x1 <- rexp(n)
x2 <- runif(n, 0, 5)
x3 <- rnorm(n)                      # irrelevant covariate
# true model: beta0(p) = 1 + qnorm(p), beta1(p) = 1 + 2 p, beta2(p) = 1
u <- runif(n)
y <- 1 + qnorm(u) + (1 + 2 * u) * x1 + x2
dat <- data.frame(y, x1, x2, x3)

fit <- piqr(y ~ x1 + x2 + x3, formula.p = ~ slp(p, 2), data = dat, nlambda = 30)
fit
best <- gof.piqr(fit, method = "BIC", plot = FALSE)
coef(fit, pos.lambda = best$posMinLambda)

# group lasso: covariates enter or leave the model as a whole
fitg <- piqr(y ~ x1 + x2 + x3, formula.p = ~ slp(p, 2), data = dat,
             nlambda = 30, penalty = "group")
plot(fitg, xvar = "lambda")

# unpenalized re-estimation of the selected model (with standard errors)

m <- refit_piqr(fit, pos.lambda = best$posMinLambda)
summary(m)

Plot a Nonlinear Parametric Quantile Model

Description

Plots the estimated coefficient functions (the columns of fun(theta, p)) with pointwise confidence bands, and a uniform QQ-plot of the fitted CDF.

Usage

## S3 method for class 'niqr'
plot(
  x,
  conf.int = TRUE,
  which = NULL,
  ask = interactive(),
  polygon = TRUE,
  ...
)

Arguments

Value

No return value, called for its side effects.

See Also

niqr()

Plot a Penalized QRCM

Description

Coefficient paths, gradient paths and estimated coefficient functions \hat\beta(p) of a piqr() fit.

Usage

## S3 method for class 'piqr'
plot(
  x,
  xvar = c("lambda", "objective", "grad", "beta"),
  pos.lambda,
  label = FALSE,
  which = NULL,
  ask = interactive(),
  polygon = TRUE,
  ...
)

Arguments

Value

No return value, called for its side effects.

See Also

piqr()

Examples

set.seed(1)
n <- 200
x1 <- runif(n); x2 <- runif(n); x3 <- runif(n)
y <- 1 + qnorm(runif(n)) + 2 * x1 - x2
fit <- piqr(y ~ x1 + x2 + x3, formula.p = ~ slp(p, 2), nlambda = 20)
oldpar <- par(mfrow = c(1, 2))
plot(fit, xvar = "lambda")
plot(fit, xvar = "grad")
par(oldpar)
plot(fit, xvar = "beta", pos.lambda = 20, which = 2)

Prediction after Nonlinear Parametric Quantile Models

Description

Prediction after Nonlinear Parametric Quantile Models

Usage

## S3 method for class 'niqr'
predict(
  object,
  type = c("beta", "CDF", "PDF", "QF", "sim"),
  newdata,
  p,
  se = TRUE,
  ...
)

Arguments

Value

• type = "beta": a list with one data frame for each column of fun, with columns p, beta, and (if se = TRUE) se, low, up (95% pointwise confidence limits).

• type = "CDF" or "PDF": a vector.

• type = "QF": a data frame with one column for each p or, if se = TRUE, a list with items fit and se.fit.

• type = "sim": a vector of simulated responses.

See Also

niqr()

Prediction after Penalized QRCM

Description

Predictions from a piqr() fit at a selected value of \lambda.

Usage

## S3 method for class 'piqr'
predict(
  object,
  pos.lambda,
  type = c("beta", "CDF", "QF", "sim"),
  newdata,
  p,
  se = !is.null(object$covar),
  lambda,
  ...
)

Arguments

Value

• type = "beta": a list with one data frame for each covariate, with columns p, beta (and se, low, up if se = TRUE).

• type = "CDF": a data frame with columns CDF and PDF.

• type = "QF": a data frame of fitted quantiles (one column for each p) or, if se = TRUE, a list with items fit and se.fit.

• type = "sim": a vector of simulated responses.

See Also

piqr()

Examples

set.seed(1)
n <- 200
x <- runif(n)
y <- qexp(runif(n)) * (1 + x)
fit <- piqr(y ~ x, formula.p = ~ slp(p, 3), nlambda = 10)
predict(fit, pos.lambda = 10, type = "QF", p = c(0.25, 0.5),
        newdata = data.frame(x = c(0, 0.5, 1)))
head(predict(fit, pos.lambda = 10, type = "CDF"))

Defunct Functions in qrcmNP

Description

These functions and interfaces were removed in 'qrcmNP' 1.0.0:

• testfit.niqr(obj, R): use test.fit() (test.fit(object, R));

• niqr(fun, fun2, x0, X, y, control): use niqr(formula, data, fun, fun2, x0) (see niqr());

• the argument display of piqr(): use verbose;

• the argument df.new of gof.piqr(): use df;

• the control names of 'qrcmNP' 0.2.x in niqr(): use niqr.control().

Usage

testfit.niqr(...)

Arguments

Value

No return value: an error is raised.

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

qrcm

plf(), slp(), test.fit()

Unpenalized Re-estimation of the Model Selected by a Penalized QRCM

Description

Following Sottile et al. (2020, Section 3.2), the penalized estimator is used to select the covariates, and the selected model is re-estimated without penalty with qrcm::iqr(), which provides asymptotic standard errors, Wald tests and goodness-of-fit diagnostics.

Usage

refit_piqr(
  object,
  pos.lambda,
  criterion = c("BIC", "AIC"),
  keep = c("covariates", "coefficients"),
  data,
  ...
)

Arguments

Value

An object of class "iqr" (see qrcm::iqr()), with an additional attribute "pos.lambda".

Note

The standard errors are conditional on the selected model and do not account for the selection step.

See Also

piqr(), gof.piqr(), qrcm::iqr(), qrcm::summary.iqr()

Examples

set.seed(1)
n <- 300
x1 <- runif(n); x2 <- runif(n)
dat <- data.frame(y = 1 + qnorm(runif(n)) + 2 * x1, x1, x2)
fit <- piqr(y ~ x1 + x2, formula.p = ~ slp(p, 2), data = dat, nlambda = 20)
m <- refit_piqr(fit)
summary(m)

Summarize a Nonlinear Parametric Quantile Model

Description

Summarize a Nonlinear Parametric Quantile Model

Usage

## S3 method for class 'niqr'
summary(object, p, ...)

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

Arguments

Value

An object of class "summary.niqr". Without p: the table of estimated parameters with standard errors, z-values and p-values, the minimized loss and the information criteria of gof.niqr(). With p: a list with one table for each value of p.

See Also

niqr(), gof.niqr()

Summarize a Penalized QRCM

Description

Summary of a piqr() fit at a selected value of \lambda.

Usage

## S3 method for class 'piqr'
summary(object, pos.lambda, SE = FALSE, p, cov = FALSE, lambda, ...)

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

Arguments

Value

An object of class "summary.piqr". Without p: a list with the selected lambda, the matrix of coefficients, their standard errors se (if requested and available), the minimized loss obj.function, the number of observations n and of nonzero parameters free.par. With p: a list with one element for each value of p, containing the estimated \beta(p) (and standard errors, if available).

See Also

piqr(), gof.piqr(), refit_piqr().

Examples

set.seed(1)
n <- 200
x1 <- runif(n); x2 <- runif(n)
y <- 1 + qnorm(runif(n)) + 2 * x1
fit <- piqr(y ~ x1 + x2, formula.p = ~ slp(p, 2), nlambda = 20)
best <- gof.piqr(fit, plot = FALSE)
summary(fit, pos.lambda = best$posMinLambda)
summary(fit, pos.lambda = best$posMinLambda, p = c(0.1, 0.5, 0.9))

Goodness-of-Fit Test for Nonlinear Parametric Quantile Models

Description

Under a correctly specified model, the fitted values of the conditional CDF, F(y_i \mid x_i, \hat\theta), are approximately uniformly distributed. This function computes the Kolmogorov-Smirnov and Cramer-von Mises statistics of their empirical distribution, and obtains p-values by parametric bootstrap, as in qrcm::test.fit().

Usage

## S3 method for class 'niqr'
test.fit(object, R = 100, trace = FALSE, ...)

Arguments

Value

A matrix with the test statistics and their bootstrap p-values.

See Also

niqr(), gof.niqr()