Package {Qest}

Type:	Package
Title:	Quantile-Based Estimators
Version:	2.0.0
Description:	Quantile-based estimators (Q-estimators) fit any parametric distribution through its quantile function, by minimizing the integrated loss of quantile regression. They are usually more robust than maximum likelihood estimators, and allow for censored and truncated data. The package implements user-defined quantile functions, built-in families (normal, gamma, continuous Poisson, uniform), linear models and proportional hazards models. The method is described in Sottile and Frumento (2022) <doi:10.1016/j.csda.2022.107471>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL:	https://github.com/gianluca-sottile/Qest
BugReports:	https://github.com/gianluca-sottile/Qest/issues
Depends:	R (≥ 4.1.0)
Imports:	graphics, matrixStats, stats, survival, 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 13:59:56 UTC; gianlucasottile
Author:	Gianluca Sottile [aut, cre], Paolo Frumento [aut]
Maintainer:	Gianluca Sottile <gianluca.sottile@unipa.it>
Repository:	CRAN
Date/Publication:	2026-10-29 18:50:02 UTC

Qest: Quantile-Based Estimators

Description

Quantile-based estimators (Q-estimators) fit a parametric model by minimizing the integrated loss of quantile regression,

\sum_i \int_0^1 w(\tau)\rho_\tau(y_i - Q(\tau \mid x_i, \theta))\,d\tau,

where Q(\tau \mid x, \theta) is the quantile function of the model (Sottile and Frumento, 2022). Q-estimators are consistent and asymptotically normal, are usually more robust than maximum likelihood estimators, and allow for censored and truncated data.

Details

The main functions are:

• Qest(): models defined by a user-written quantile function, or by one of the built-in families Qnorm, Qgamma, Qpois and Qunif;

• Qlm(): linear models with normal errors (closed-form estimating equations);

• Qcoxph(): proportional hazards models with a piecewise linear baseline cumulative hazard.

The weighting function w(\tau) (e.g., wtrunc()) can be used to increase the robustness of the estimators.

Author(s)

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

Authors:

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

• Paolo Frumento paolo.frumento@unipi.it

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

Useful links:

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

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

Q-Estimation of Proportional Hazards Regression Models

Description

Fits a proportional hazards model with the Q-estimator of Sottile and Frumento (2022). The baseline cumulative hazard H_0(t) is piecewise linear, so that the model is fully parametric, and Q(\tau \mid x) = H_0^{-1}(-\exp(-x^\top\beta)\log(1 - \tau)).

Usage

Qcoxph(
  formula,
  weights,
  start,
  data,
  subset,
  na.action,
  knots,
  wtau = NULL,
  control = Qcoxph.control(),
  ...
)

Arguments

Details

The model is h(t \mid x) = h_0(t)\exp(x^\top\beta), with H_0(t) = \sum_l \exp(\gamma_l)\,b_l(t), where b_l is a piecewise linear basis. All quantities of the estimating equations have closed forms (Appendix D2 of the paper). The coefficients have the same interpretation as in survival::coxph(); the estimates are typically very close, but more robust. Starting points are obtained from the partial likelihood and the Breslow estimator of H_0.

Methods are available to compute the fitted survival curves (survfit.Qcoxph(), which returns an object of class "survfit"), predictions (predict.Qcoxph()), residuals (residuals.Qcoxph()) and the baseline cumulative hazard (basehaz.Qcoxph()). All of them are based on the fitted parametric model.

Value

An object of class c("Qcoxph", "Qest"), a list with items

and terms, assign, formula, call, y, x, model, xlevels, na.action.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

survfit.Qcoxph(), predict.Qcoxph(), residuals.Qcoxph(), basehaz.Qcoxph(), Qest()

Examples

library(survival)
set.seed(1)
n <- 200
x <- runif(n, 0, 3)
t <- rweibull(n, shape = 2, scale = (1 / exp(2 + 2 * x))^(1 / 2))
cens <- runif(n, 0, 1)
dat <- data.frame(y = pmin(t, cens), d = as.numeric(t <= cens), x = x)

m1 <- coxph(Surv(y, d) ~ x, data = dat)
m2 <- Qcoxph(Surv(y, d) ~ x, data = dat)
summary(m2)
cbind(coxph = coef(m1), Qcoxph = coef(m2))
sf <- survfit(m2, newdata = data.frame(x = c(0.5, 1.5)))
plot(sf, col = 1:2)

Quantile-Based Estimators (Q-Estimators)

Description

Fits a parametric model by minimizing the integrated loss of quantile regression (Sottile and Frumento, 2022). The model is specified through its quantile function Q(\tau \mid x, \theta), either written by the user or chosen among the built-in families (see Qfamily). Censored and truncated data are allowed.

Usage

Qest(
  formula,
  Q,
  weights,
  start,
  data,
  subset,
  na.action,
  ntau = 199,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Arguments

Details

The Q-estimator minimizes

L(\theta) = \sum_i \int_0^1 w(\tau)\rho_\tau(y_i - Q(\tau \mid x_i, \theta))\,d\tau,

where \rho_\tau(u) = u(\tau - I(u < 0)). It is consistent and asymptotically normal, and its robustness can be tuned by choosing w(\tau) (Sottile and Frumento, 2022). For censored and truncated data, the estimating equations of Appendix A of the paper are solved.

User-defined quantile functions. Q(theta, tau, data) must have these three arguments, in this order (other arguments can follow and are passed through ...), and must return, for each row of data, Q(\tau \mid x_i, \theta). The argument tau is either a matrix with one row for each row of data (then Q returns a matrix of the same size), or a vector with one value for each row of data (then Q returns a vector). data is the data frame used to fit the model (after removing missing values, subset and observations with zero weight). Q must be increasing in tau and twice differentiable in theta; it is evaluated on a grid of ntau quantile orders in [5 \cdot 10^{-4}, 1 - 5 \cdot 10^{-4}], and its derivatives with respect to theta are computed by central differences.

Starting points. If start contains NA (or control$restart = TRUE), a nonparametric estimate \hat F(y \mid x) is obtained from a piecewise-constant hazard model that uses the covariates in formula, and the starting points minimize \sum_i (y_i - Q(\hat F(y_i \mid x_i) \mid x_i, \theta))^2 (Sottile and Frumento, 2022, Algorithm 1), searched from the values in start (with NA replaced by zero): with poorly scaled data, rough starting values for all parameters are recommended.

Algorithm. The estimating equations are solved by Newton-Raphson with backtracking, switching to Levenberg-Marquardt steps when the Newton step fails. The Jacobian is the exact derivative of the discretized estimating equations. Standard errors are computed with the sandwich formula.

Value

An object of class "Qest", a list with items

and the usual call, formula, terms, data, na.action, xlevels, contrasts, control.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

Qfamily, Qlm(), Qcoxph(), summary.Qest(), wtrunc(), invQ().

Examples

# Normal linear model
Qlinmod <- function(theta, tau, data) {
  X <- model.matrix(~ x1 + x2, data = data)
  qnorm(tau, c(X %*% theta[-1]), exp(theta[1]))
}
set.seed(1)
n <- 100
dat <- data.frame(x1 = rnorm(n), x2 = runif(n, 0, 3))
dat$y <- 4 + dat$x1 + 2 * dat$x2 + rnorm(n)
m1 <- Qest(y ~ x1 + x2, Q = Qlinmod, start = rep(NA, 4), data = dat)
m2 <- Qest(y ~ x1 + x2, Q = Qnorm, data = dat)    # the same model, as a Qfamily
m3 <- Qlm(y ~ x1 + x2, data = dat)                # ... or with Qlm (fastest)
summary(m1)
cbind(user = coef(m1), family = coef(m2))


# A Weibull proportional hazards model with censored data
QWeibPH <- function(theta, tau, data) {
  shape <- exp(theta[1])
  X <- model.matrix(~ x, data = data)
  qweibull(tau, shape = shape, scale = exp(-c(X %*% theta[-1]) / shape))
}
dat <- data.frame(x = runif(n, 0, 3))
t <- QWeibPH(c(log(2), -0.5, 1), runif(n), dat)
cens <- runif(n, 0, 2)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
m4 <- Qest(survival::Surv(y, d) ~ x, Q = QWeibPH, start = rep(NA, 3), data = dat)
summary(m4)

# Gamma model with more weight around the median
y <- rgamma(n, shape = 3, scale = 2)
m5 <- Qest(y ~ 1, Q = Qgamma, wtau = function(tau, h) dnorm((tau - 0.5) / h), h = 0.2)
coef(m5)

Control Parameters of Qest, Qlm and Qcoxph

Description

Control Parameters of Qest, Qlm and Qcoxph

Usage

Qest.control(
  tol = 1e-08,
  maxit = NULL,
  display = FALSE,
  restart = FALSE,
  deriv.step = 1e-04,
  ...
)

Qcoxph.control(tol = 1e-08, maxit = NULL, display = FALSE, ...)

Arguments

Value

A list with the control parameters.

See Also

Qest(), Qlm(), Qcoxph()

Examples

Qest.control(tol = 1e-10, display = TRUE)

Family Objects for Qest

Description

Built-in parametric models for Qest(), with analytical (or nearly analytical) estimating equations: much faster and more accurate than the equivalent user-defined quantile functions.

Usage

Qgamma()

Qnorm()

Qpois(offset = NULL)

Qunif(min = TRUE)

Arguments

Details

A family can be passed to Qest() by name ("Qpois"), as a function (Qpois) or as a call (Qpois()); the latter allows to pass arguments.

• Qnorm: normal homoskedastic model, Q(\tau \mid x) = x^\top\beta + \sigma\Phi^{-1}(\tau); parameters log(sigma), beta. With complete data, Qlm() fits the same model with a richer output.

• Qgamma: Gamma model with shape a and log-scale x^\top\beta; parameters log(shape), beta.

• Qpois: Poisson model with log-rate x^\top\beta. To obtain a continuous quantile function, qpois is replaced by the inverse with respect to y of the regularized upper incomplete gamma function \Gamma(y, \lambda)/\Gamma(y) (Ilienko, 2013). The model should be applied to a jittered response, y + runif(n).

• Qunif: Uniform model U(a(x), b(x)) with a(x) = x^\top\alpha and b(x) = x^\top\beta, or U(0, b(x)) if min = FALSE.

Qnorm and Qgamma can be used with censored and truncated data; Qpois and Qunif cannot. Qgamma and Qpois require an intercept.

Value

An object of class "Qfamily".

References

Ilienko, A. (2013). Continuous counterparts of Poisson and binomial distributions and their properties. Annales Univ. Sci. Budapest., Sect. Comp., 39, 137-147.

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

Qest()

Examples

set.seed(1)
n <- 250
x <- runif(n)
eta <- 1 + 2 * x

m1 <- Qest(y ~ x, Qnorm, data = data.frame(y = rnorm(n, eta, exp(1)), x = x))
m2 <- Qest(y ~ x, Qgamma, data = data.frame(y = rgamma(n, shape = exp(1), scale = exp(eta)), x = x))
m3 <- Qest(y ~ x, Qpois, data = data.frame(y = rpois(n, exp(eta)) + runif(n), x = x))
m4 <- Qest(y ~ x, Qunif(min = TRUE), data = data.frame(y = runif(n, 0, eta), x = x))
m5 <- Qest(y ~ x, Qunif(min = FALSE), data = data.frame(y = runif(n, 0, eta), x = x))
coef(m2)

Q-Estimation of Linear Regression Models

Description

Fits a linear model with normal homoskedastic errors, Q(\tau \mid x) = x^\top\beta + \sigma\Phi^{-1}(\tau), with the Q-estimator of Sottile and Frumento (2022). The estimating equations and the Hessian have closed forms (Appendix D1 of the paper), and the estimator is fast and robust to outliers in the response.

Usage

Qlm(
  formula,
  data,
  subset,
  weights,
  na.action,
  start = NULL,
  contrasts = NULL,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Qlm.fit(
  y,
  X,
  w = rep(1, nobs),
  start = NULL,
  wtau = NULL,
  control = Qest.control(),
  ...
)

Arguments

Details

Qlm(formula) estimates the same model as Qest(formula, Q = Qnorm), with a much richer output. Censored and truncated data are not allowed (use Qest() with Q = Qnorm). Standard errors are computed with the sandwich formula, and the tests in summary.Qlm() are based on the asymptotic normal distribution.

Value

Qlm returns an object of class c("Qlm", "Qest") with items

and rank, df.residual, weights, converged, n.it, call, terms, model, xlevels, contrasts, na.action. Qlm.fit returns a similar list.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

See Also

summary.Qlm(), predict.Qlm(), Qest()

Examples

set.seed(1234)
n <- 100
x1 <- rnorm(n)
x2 <- runif(n, 0, 3)
y <- rnorm(n, 4 + x1 + 2 * x2, 1)
y[1:5] <- y[1:5] + 20      # outliers

m1 <- Qlm(y ~ x1 + x2)
summary(m1)
m2 <- Qlm(y ~ x1 + x2, wtau = wtrunc, delta.left = 0.05, delta.right = 0.05)
cbind(lm = coef(lm(y ~ x1 + x2)), Qlm = coef(m1), Qlm.trunc = coef(m2))

Baseline Cumulative Hazard of a Qcoxph Model

Description

Baseline Cumulative Hazard of a Qcoxph Model

Usage

basehaz.Qcoxph(fit, centered = TRUE, se.fit = FALSE, times)

Arguments

Details

The baseline cumulative hazard of the fitted model is piecewise linear, H_0(t) = \sum_l \exp(\gamma_l) b_l(t). Unlike survival::basehaz(), this is not a step function.

Value

A data frame with columns hazard (the cumulative hazard), time and, if se.fit = TRUE, varhaz.

See Also

Qcoxph(), survfit.Qcoxph()

Examples

library(survival)
set.seed(1)
x <- runif(200)
t <- rexp(200, exp(x))
cens <- runif(200, 0, 3)
fit <- Qcoxph(Surv(pmin(t, cens), t <= cens) ~ x)
head(basehaz.Qcoxph(fit, se.fit = TRUE))

Inverse of a Quantile Function

Description

Given a quantile function Q(\tau \mid x, \theta) and a value of \theta, computes F(y_i \mid x_i, \theta) by bisection.

Usage

invQ(Q, theta, y, data, n.it = 17)

Arguments

Value

A vector with F(y_i \mid x_i, \theta).

See Also

Qest()

Examples

Qlinmod <- function(theta, tau, data) {
  qnorm(tau, theta[2] + theta[3] * data$x, exp(theta[1]))
}
set.seed(1)
dat <- data.frame(x = runif(10))
U <- runif(10)
y <- Qlinmod(c(0, 1, 2), U, dat)
cbind(U, invQ(Qlinmod, c(0, 1, 2), y, dat))

Diagnostic Plot of Q-Estimators

Description

If the model is correctly specified, F(y_i \mid x_i) are a sample from a U(0, 1) distribution. The plot compares their empirical distribution (the Kaplan-Meier estimator, with censored or truncated data) with the uniform distribution function.

Usage

## S3 method for class 'Qest'
plot(x, conf.int = TRUE, ...)

Arguments

Value

Invisibly, a data frame with the Kaplan-Meier estimate of the distribution of F(y_i \mid x_i).

See Also

Qest()

Examples

set.seed(1)
y <- rgamma(300, shape = 2, scale = 3)
plot(Qest(y ~ 1, Q = Qgamma))   # good fit
plot(Qest(y ~ 1, Q = Qnorm))    # poor fit

Predictions from a Qcoxph Model

Description

Predictions from a Qcoxph Model

Usage

## S3 method for class 'Qcoxph'
predict(
  object,
  newdata,
  type = c("lp", "risk", "expected", "survival", "quantile"),
  se.fit = FALSE,
  p = c(0.25, 0.5, 0.75),
  ...
)

Arguments

Value

A vector, or a list with items fit and se.fit; for type = "quantile", a matrix with one column for each value of p.

See Also

Qcoxph(), survfit.Qcoxph()

Examples

library(survival)
set.seed(1)
dat <- data.frame(x = runif(200))
t <- rexp(200, exp(dat$x)); cens <- runif(200, 0, 3)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
fit <- Qcoxph(Surv(y, d) ~ x, data = dat)
predict(fit, newdata = data.frame(x = c(0, 0.5, 1)), type = "risk", se.fit = TRUE)
predict(fit, newdata = data.frame(x = c(0, 1)), type = "quantile", p = c(0.25, 0.5))

Predictions from Q-Estimators

Description

Computes the fitted quantiles Q(p \mid x), or the fitted distribution function F(y \mid x) at the observed (or new) values of the response.

Usage

## S3 method for class 'Qest'
predict(
  object,
  newdata,
  type = c("quantile", "CDF"),
  p = c(0.1, 0.25, 0.5, 0.75, 0.9),
  ...
)

Arguments

Details

For a user-defined Q, newdata is passed as the argument data of Q. For type = "CDF", F(y \mid x) is computed by inverting Q (see invQ()). For censored data, the CDF is evaluated at the observed times.

Value

For type = "quantile", a matrix with one row for each observation and one column for each value of p; for type = "CDF", a vector.

See Also

Qest(), invQ()

Examples

set.seed(1)
dat <- data.frame(x = runif(200))
dat$y <- rgamma(200, shape = 2, scale = exp(1 + dat$x))
fit <- Qest(y ~ x, Q = Qgamma, data = dat)
predict(fit, newdata = data.frame(x = c(0, 1)), p = c(0.1, 0.5, 0.9))
head(predict(fit, type = "CDF"))

Predictions from Qlm Models

Description

Predictions from Qlm Models

Usage

## S3 method for class 'Qlm'
predict(
  object,
  newdata,
  type = c("response", "quantile"),
  se.fit = FALSE,
  p = c(0.1, 0.25, 0.5, 0.75, 0.9),
  na.action = stats::na.pass,
  ...
)

Arguments

Value

A vector, a list with items fit and se.fit, or (for type = "quantile") a matrix with one column for each value of p.

See Also

Qlm()

Examples

set.seed(1)
x <- runif(100)
y <- 1 + 2 * x + rnorm(100)
fit <- Qlm(y ~ x)
predict(fit, newdata = data.frame(x = c(0, 0.5)), se.fit = TRUE)
predict(fit, newdata = data.frame(x = c(0, 0.5)), type = "quantile", p = c(0.1, 0.9))

Residuals of a Qcoxph Model

Description

Residuals of a Qcoxph Model

Usage

## S3 method for class 'Qcoxph'
residuals(object, type = c("martingale", "deviance"), ...)

Arguments

Value

A vector of residuals.

See Also

Qcoxph()

Summarizing Qcoxph Fits

Description

Summarizing Qcoxph Fits

Usage

## S3 method for class 'Qcoxph'
summary(object, conf.int = 0.95, scale = 1, ...)

## S3 method for class 'summary.Qcoxph'
print(
  x,
  digits = max(getOption("digits") - 3, 3),
  signif.stars = getOption("show.signif.stars"),
  ...
)

Arguments

Value

An object of class "summary.Qcoxph", with the table of coefficients (coefficients), the confidence intervals of the hazard ratios (conf.int), the concordance index and its standard error (concordance), and the loss (obj.function).

See Also

Qcoxph()

Summarizing Q-Estimators

Description

Summary methods for objects of class "Qest" (fitted by Qest()) and "Qlm" (fitted by Qlm()).

Usage

## S3 method for class 'Qest'
summary(object, covar = FALSE, ...)

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

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

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

Arguments

Details

The tests are based on the asymptotic normal distribution of the estimators, with standard errors from the sandwich formula (Sottile and Frumento, 2022).

Value

summary.Qest returns an object of class "summary.Qest" with the table of coefficients (coefficients), the loss (obj.function), the numbers of observations and parameters, the convergence status and, optionally, the covariance matrix. summary.Qlm returns an object of class "summary.Qlm" with the table of coefficients, the estimate of \sigma and its standard error, the residuals and, optionally, the correlation matrix.

See Also

Qest(), Qlm()

Examples

set.seed(1)
y <- rgamma(200, shape = 3, scale = 2)
fit <- Qest(y ~ 1, Q = Qgamma)
summary(fit, covar = TRUE)

Survival Curves from a Qcoxph Model

Description

Computes the survival curves S(t \mid x) = \exp\{-H_0(t)\exp(x^\top\beta)\} of the fitted model, with pointwise confidence intervals based on the delta method.

Usage

## S3 method for class 'Qcoxph'
survfit(
  formula,
  newdata,
  se.fit = TRUE,
  conf.int = 0.95,
  conf.type = c("log", "log-log", "plain", "none"),
  times,
  ...
)

Arguments

Value

An object of class "survfit" (see survival::survfit.object): the methods print, plot, lines and summary of 'survival' can be used.

See Also

Qcoxph(), predict.Qcoxph()

Examples

library(survival)
set.seed(1)
dat <- data.frame(x = runif(200))
t <- rexp(200, exp(dat$x)); cens <- runif(200, 0, 3)
dat$y <- pmin(t, cens); dat$d <- as.numeric(t <= cens)
fit <- Qcoxph(Surv(y, d) ~ x, data = dat)
sf <- survfit(fit, newdata = data.frame(x = c(0, 1)))
plot(sf, col = 1:2, conf.int = TRUE)

Weights for Truncated Quantile Integration

Description

A weighting function w(\tau) that assigns zero weight to the quantiles below delta.left and above 1 - delta.right, to be used as the argument wtau of Qest(), Qlm() and Qcoxph(). Truncated weights make the estimators more robust to outliers (Sottile and Frumento, 2022, Section 3).

Usage

wtrunc(
  tau,
  delta.left = 0.05,
  delta.right = 0.05,
  smooth = FALSE,
  sigma = 0.01
)

Arguments

Details

With smooth = FALSE, w(\tau) = I(\delta_L \le \tau \le 1 - \delta_R). With smooth = TRUE, the indicators are replaced by \Phi((\tau - \delta_L)/\sigma) and 1 - \Phi((\tau - 1 + \delta_R)/\sigma). A delta equal to zero means no truncation on that side. The arguments delta.left, delta.right, smooth and sigma are passed through the ... of the fitting functions.

Value

A vector of weights of the same length as tau.

References

Sottile, G. and Frumento, P. (2022). Robust estimation and regression with parametric quantile functions. Computational Statistics and Data Analysis, 171, 107471. doi:10.1016/j.csda.2022.107471

Examples

taus <- seq(0, 1, length.out = 1000)
plot(taus, wtrunc(taus, delta.left = 0.05, delta.right = 0.1), type = "l")
lines(taus, wtrunc(taus, 0.05, 0.1, smooth = TRUE, sigma = 0.01), col = 2)

# in a model
set.seed(1)
y <- c(rnorm(95), rnorm(5, 10))   # 5% of outliers
fit <- Qlm(y ~ 1, wtau = wtrunc, delta.left = 0.05, delta.right = 0.05)
coef(fit)