Package 'EL' reference manual

Title:	Two-Sample Empirical Likelihood
Description:	Empirical likelihood (EL) inference for two-sample problems. The following statistics are included: the difference of two-sample means, smooth Huber estimators, quantile (qdiff) and cumulative distribution functions (fdiff), probability-probability (P-P) and quantile-quantile (Q-Q) plots as well as receiver operating characteristic (ROC) curves. Also includes two-sample block-wise empirical likelihood (BEL) and a frequency-domain empirical likelihood test for autocorrelation differences (FDEL). Methods for EL, P-P, Q-Q, ROC, qdiff and fdiff are based on Valeinis and Cers (2011) <http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf>.
Authors:	Janis Valeinis [aut] (ORCID: <https://orcid.org/0000-0003-0989-0444>), Edmunds Cers [aut], Janis Gredzens [cre] (ORCID: <https://orcid.org/0009-0009-4890-4897>), Reinis Alksnis [ctb] (ORCID: <https://orcid.org/0000-0002-5512-5463>)
Maintainer:	Janis Gredzens <[email protected]>
License:	GPL (>= 2)
Version:	1.4
Built:	2026-05-12 21:21:58 UTC
Source:	https://github.com/cran/EL

The two-sample blockwise empirical likelihood statistic for differences in means

Description

Calculates blockwise empirical likelihood test for the difference of two sample means.

Usage

BEL.means(X, Y, M_1, M_2, Delta = 0)
BEL.means(X, Y, M_1, M_2, Delta = 0)

Arguments

X, Y

vectors of data values.

M_1, M_2

positive integers specifying block length for X and Y, respectively.

Delta

hypothesized difference of two populations.

Value

A list of class "htest" containing following components: method - the character string of the test. data.name - a character string with the names of the input data. Delta0 - the specified hypothesized value of mean differences under the null hypothesis statistic - the value of the test statistic. p.value - the p-value for the test.

Author(s)

R. Alksnis, J. Valeinis

Examples

# Basic example
Delta0 <- 1.5
X <- arima.sim(n = 400, model = list(ar = .3))
Y <- arima.sim(n = 400, model = list(ar = .5)) + Delta0
BEL.means(X, Y, 10, 20, Delta = Delta0)

# Basic example
Delta0 <- 1.5
X <- arima.sim(n = 400, model = list(ar = .3))
Y <- arima.sim(n = 400, model = list(ar = .5)) + Delta0
BEL.means(X, Y, 10, 20, Delta = Delta0)

Empirical likelihood test for the difference of smoothed Huber estimators

Description

Empirical likelihood inference for the difference of smoothed Huber estimators. This includes a test for the null hypothesis of a constant difference of smoothed Huber estimators, a confidence interval, and the EL estimator.

Usage

EL.Huber(
  X,
  Y,
  mu = 0,
  conf.level = 0.95,
  scaleX = 1,
  scaleY = 1,
  VX = 2.046,
  VY = 2.046,
  k = 1.35
)
EL.Huber(
  X,
  Y,
  mu = 0,
  conf.level = 0.95,
  scaleX = 1,
  scaleY = 1,
  VX = 2.046,
  VY = 2.046,
  k = 1.35
)

Arguments

X

A numeric vector of data values.

Y

A numeric vector of data values.

mu

A number specifying the null hypothesis value for the difference. Default is 0.

conf.level

Confidence level for the reported confidence interval (default 0.95).

scaleX

The scale estimate of sample X. Default is 1.

scaleY

The scale estimate of sample Y. Default is 1.

VX

The asymptotic variance of the initial (nonsmooth) Huber estimator for sample X. Default is 2.046.

VY

The asymptotic variance of the initial (nonsmooth) Huber estimator for sample Y. Default is 2.046.

k

Tuning parameter for the Huber estimator. Default is 1.35.

Details

A common choice for a robust scale estimate (parameters scaleX and scaleY) is the median absolute deviation (MAD).

Value

A list of class "htest" with components:

estimate: The empirical likelihood estimate for the difference of two smoothed Huber estimators.
conf.int: A confidence interval for the difference of two smoothed Huber estimators.
p.value: The p-value for the test.
statistic: The value of the test statistic.
method: The character string "Empirical likelihood smoothed Huber estimator difference test".
null.value: The hypothesised difference mu under $H_0$ .
data.name: A character string giving the names of the data.

Author(s)

E. Cers, J. Valeinis

References

Valeinis, J. and Cers, E. Extending the two-sample empirical likelihood. Preprint: http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf

Hampel, F., Hennig, C. and Ronchetti, E. A. (2011). A smoothing principle for the Huber and other location M-estimators. Computational Statistics & Data Analysis, 55(1), 324–337.

Examples

X <- rnorm(100)
Y <- rnorm(100)
t.test(X, Y)
EL.means(X, Y)
EL.Huber(X, Y)

X <- rnorm(100)
Y <- rnorm(100)
t.test(X, Y)
EL.means(X, Y)
EL.Huber(X, Y)

Empirical likelihood test for the difference of two sample means

Description

Empirical likelihood inference for the difference of two sample means. This includes a test for the null hypothesis of a constant mean difference, a confidence interval, and the EL estimator.

Usage

EL.means(X, Y, mu = 0, conf.level = 0.95)
EL.means(X, Y, mu = 0, conf.level = 0.95)

Arguments

X

A numeric vector of data values.

Y

A numeric vector of data values.

mu

A number specifying the null hypothesis value for the mean difference. Default is 0.

conf.level

Confidence level for the reported confidence interval (default 0.95).

Value

A list of class "htest" with components:

estimate: The empirical likelihood estimate of the mean difference.
conf.int: A confidence interval for the mean difference.
p.value: The p-value for the test.
statistic: The value of the test statistic.
method: The character string "Empirical likelihood mean difference test".
null.value: The hypothesised mean difference mu under $H_0$ .
data.name: A character string giving the names of the data.

Author(s)

E. Cers, J. Valeinis

References

Valeinis, J., Cers, E. and Cielens, J. (2010). Two-sample problems in statistical data modelling. Mathematical Modelling and Analysis, 15(1), 137–151.

Valeinis, J. and Cers, E. Extending the two-sample empirical likelihood. Preprint: http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf

Examples

X <- rnorm(100)
Y <- rnorm(100)
t.test(X, Y)
EL.means(X, Y)
EL.Huber(X, Y)

X <- rnorm(100)
Y <- rnorm(100)
t.test(X, Y)
EL.means(X, Y)
EL.Huber(X, Y)

Draws plots using the smoothed two-sample empirical likelihood method

Description

Draws P-P and Q-Q plots, ROC curves, quantile differences (qdiff) and CDF differences (ddiff) and their respective confidence bands (pointwise or simultaneous) using the empirical likelihood method.

Usage

EL.plot(
  method,
  X,
  Y,
  bw = bw.nrd0,
  conf.level = NULL,
  simultaneous = FALSE,
  bootstrap.samples = 300,
  more.warnings = FALSE,
  ...
)
EL.plot(
  method,
  X,
  Y,
  bw = bw.nrd0,
  conf.level = NULL,
  simultaneous = FALSE,
  bootstrap.samples = 300,
  more.warnings = FALSE,
  ...
)

Arguments

method

One of "pp", "qq", "roc", "qdiff", or "fdiff".

X

A numeric vector of data values.

Y

A numeric vector of data values.

bw

A function taking a numeric vector and returning a bandwidth, or a numeric vector of length two giving the bandwidths for X and Y respectively. Default is bw.nrd0.

conf.level

Confidence level for the intervals, a number in $(0,1)$ , or NULL to skip confidence band calculation.

simultaneous

Logical. If TRUE, simultaneous confidence bands are constructed via a nonparametric bootstrap. Default is FALSE (pointwise intervals).

bootstrap.samples

Integer. Number of bootstrap samples used when simultaneous = TRUE. Default is 300.

more.warnings

Logical. If FALSE (default), a single warning is issued if any problem occurs. If TRUE, a warning is produced for every point with a problem.

...

Further arguments passed to the plot function.

Details

The plotting interval for P-P plots, ROC curves and differences of quantile functions is $[0, 1]$ . The Q-Q plot is drawn from the minimum to the maximum of Y. For the plot of distribution function differences the interval from $\max(\min(X), \min(Y))$ to $\min(\max(X), \max(Y))$ is used.

Confidence bands are drawn only if conf.level is not NULL.

When constructing simultaneous confidence bands, the plot is drawn on an interval narrowed by 5% on both sides, since the procedure is usually sensitive at the endpoints. The confidence level is bootstrapped using 50 evenly spaced points in this interval. If the default interval produces excessively wide bands, use EL.smooth where intervals are specified manually. Note that calculation of simultaneous confidence bands can be slow.

Value

A ggplot object. The plot can be further customized using ggplot2 functions, as shown in the examples.

Author(s)

E. Cers, J. Valeinis

References

Valeinis, J. and Cers, E. Extending the two-sample empirical likelihood. Preprint: http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf

Hall, P. and Owen, A. (1993). Empirical likelihood bands in density estimation. Journal of Computational and Graphical Statistics, 2(3), 273–289.

Examples

## The examples showcase all available graphs
set.seed(42)
X1 <- rnorm(100, 0.5, 1.5)
X2 <- rnorm(100, 0, 1)


xlim <- c(min(X1, X2) - 0.5, max(X1, X2) + 0.5)
D1 <- density(X1)
D2 <- density(X2)
df <- data.frame(x1 = D1$x, y1 = D1$y, x2 = D2$x, y2 = D2$y)
p1 <- ggplot2::ggplot(data = df) +
    ggplot2::geom_line(ggplot2::aes(x = x2, y = y2,
        color = paste0('X2 (bw=', round(D2$bw, 2), ')'))) +
    ggplot2::geom_line(ggplot2::aes(x = x1, y = y1,
        color = paste0('X1 (bw=', round(D1$bw, 2), ')'))) +
    ggplot2::guides(color = ggplot2::guide_legend(title = NULL)) +
    ggplot2::theme_minimal() +
    ggplot2::theme(legend.position = "top") +
    ggplot2::labs(x = "X", y = "Density")
p1

# CDF differences
p2 <- EL.plot("fdiff", X1, X2, main = "F difference", conf.level = 0.95)
tt <- seq(max(c(min(X1), min(X2))), min(c(max(X1), max(X2))), length = 30)
ee <- ecdf(X2)(tt) - ecdf(X1)(tt)
p2 <- p2 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p2

# Quantile differences
p3 <- EL.plot("qdiff", X1, X2, main = "Quantile difference", conf.level = 0.95)
tt <- seq(0.01, 0.99, length = 30)
ee <- quantile(X2, tt) - quantile(X1, tt)
p3 <- p3 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p3

# Q-Q plot
p4 <- EL.plot("qq", X1, X2, main = "Q-Q plot", conf.level = 0.95)
tt <- seq(min(X2), max(X2), length = 30)
ee <- quantile(X1, ecdf(X2)(tt))
p4 <- p4 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p4

# P-P plot
p5 <- EL.plot("pp", X1, X2, main = "P-P plot", conf.level = 0.95, ylim = c(0, 1))
tt <- seq(0.01, 0.99, length = 30)
ee <- ecdf(X1)(quantile(X2, tt))
p5 <- p5 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p5

# ROC curve
p6 <- EL.plot("roc", X1, X2, main = "ROC curve", conf.level = 0.95, ylim = c(0, 1))
tt <- seq(0.01, 0.99, length = 30)
ee <- 1 - ecdf(X1)(quantile(X2, 1 - tt))
p6 <- p6 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p6

# To show all plots at once:
# require(cowplot)
# cowplot::plot_grid(p1, p2, p3, p4, p5, p6, ncol = 2)


## The examples showcase all available graphs
set.seed(42)
X1 <- rnorm(100, 0.5, 1.5)
X2 <- rnorm(100, 0, 1)


xlim <- c(min(X1, X2) - 0.5, max(X1, X2) + 0.5)
D1 <- density(X1)
D2 <- density(X2)
df <- data.frame(x1 = D1$x, y1 = D1$y, x2 = D2$x, y2 = D2$y)
p1 <- ggplot2::ggplot(data = df) +
    ggplot2::geom_line(ggplot2::aes(x = x2, y = y2,
        color = paste0('X2 (bw=', round(D2$bw, 2), ')'))) +
    ggplot2::geom_line(ggplot2::aes(x = x1, y = y1,
        color = paste0('X1 (bw=', round(D1$bw, 2), ')'))) +
    ggplot2::guides(color = ggplot2::guide_legend(title = NULL)) +
    ggplot2::theme_minimal() +
    ggplot2::theme(legend.position = "top") +
    ggplot2::labs(x = "X", y = "Density")
p1

# CDF differences
p2 <- EL.plot("fdiff", X1, X2, main = "F difference", conf.level = 0.95)
tt <- seq(max(c(min(X1), min(X2))), min(c(max(X1), max(X2))), length = 30)
ee <- ecdf(X2)(tt) - ecdf(X1)(tt)
p2 <- p2 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p2

# Quantile differences
p3 <- EL.plot("qdiff", X1, X2, main = "Quantile difference", conf.level = 0.95)
tt <- seq(0.01, 0.99, length = 30)
ee <- quantile(X2, tt) - quantile(X1, tt)
p3 <- p3 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p3

# Q-Q plot
p4 <- EL.plot("qq", X1, X2, main = "Q-Q plot", conf.level = 0.95)
tt <- seq(min(X2), max(X2), length = 30)
ee <- quantile(X1, ecdf(X2)(tt))
p4 <- p4 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p4

# P-P plot
p5 <- EL.plot("pp", X1, X2, main = "P-P plot", conf.level = 0.95, ylim = c(0, 1))
tt <- seq(0.01, 0.99, length = 30)
ee <- ecdf(X1)(quantile(X2, tt))
p5 <- p5 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p5

# ROC curve
p6 <- EL.plot("roc", X1, X2, main = "ROC curve", conf.level = 0.95, ylim = c(0, 1))
tt <- seq(0.01, 0.99, length = 30)
ee <- 1 - ecdf(X1)(quantile(X2, 1 - tt))
p6 <- p6 + ggplot2::geom_point(data = data.frame(tt = tt, ee = ee),
                                ggplot2::aes(x = tt, y = ee))
p6

# To show all plots at once:
# require(cowplot)
# cowplot::plot_grid(p1, p2, p3, p4, p5, p6, ncol = 2)

Smooth estimates and confidence intervals using the smoothed two-sample EL method

Description

Calculates estimates and pointwise confidence intervals (or simultaneous bands) for P-P and Q-Q plots, ROC curves, quantile differences (qdiff) and CDF differences (ddiff) using the smoothed empirical likelihood method.

Usage

EL.smooth(
  method,
  X,
  Y,
  t,
  bw = bw.nrd0,
  conf.level = NULL,
  simultaneous = FALSE,
  bootstrap.samples = 300,
  more.warnings = FALSE
)
EL.smooth(
  method,
  X,
  Y,
  t,
  bw = bw.nrd0,
  conf.level = NULL,
  simultaneous = FALSE,
  bootstrap.samples = 300,
  more.warnings = FALSE
)

Arguments

method

One of "pp", "qq", "roc", "qdiff", or "fdiff".

X

A numeric vector of data values.

Y

A numeric vector of data values.

t

A numeric vector of points at which to calculate estimates and confidence intervals.

bw

A function taking a numeric vector and returning a bandwidth, or a numeric vector of length two giving the bandwidths for X and Y respectively. Default is bw.nrd0.

conf.level

Confidence level for the intervals, a number in $(0,1)$ , or NULL to skip confidence band calculation.

simultaneous

Logical. If TRUE, simultaneous confidence bands are constructed via a nonparametric bootstrap. Default is FALSE (pointwise intervals).

bootstrap.samples

Integer. Number of bootstrap samples used when simultaneous = TRUE. Default is 300.

more.warnings

Logical. If FALSE (default), a single warning is issued if any problem occurs. If TRUE, a warning is produced for every point with a problem.

Details

Confidence bands are drawn only if conf.level is not NULL.

When constructing simultaneous confidence bands, check that the chosen range of t values does not produce excessively wide bands (for example, for a P-P plot the interval $[0.05, 0.95]$ is typically a sensible choice). This should be verified for each dataset. Note that simultaneous band calculation can be slow.

Value

A list with components:

estimate: Estimated values at points t.
conf.int: A two-row matrix where each column gives the lower and upper confidence bounds at the corresponding point in t.
simultaneous.conf.int: Logical; TRUE if simultaneous bands were constructed.
bootstrap.crit: The bootstrap critical value of the $-2\log$ -likelihood statistic for simultaneous bands at level conf.level. Only present when conf.level is not NULL and simultaneous = TRUE.

Author(s)

E. Cers, J. Valeinis

References

Valeinis, J. and Cers, E. Extending the two-sample empirical likelihood. Preprint: http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf

Hall, P. and Owen, A. (1993). Empirical likelihood bands in density estimation. Journal of Computational and Graphical Statistics, 2(3), 273–289.

Examples


#### Simultaneous confidence bands for a P-P plot
X1 <- rnorm(200)
X2 <- rnorm(200, 1)
x <- seq(0.05, 0.95, length = 19)
y <- EL.smooth("pp", X1, X2, x, conf.level = 0.95,
               simultaneous = TRUE, bw = c(0.3, 0.3))
conf.int <- data.frame(x = x, ci.l = y$conf.int[1,], ci.u = y$conf.int[2,])

## Plot with both pointwise and simultaneous confidence bands
EL.plot("pp", X1, X2, conf.level = 0.95, bw = c(0.3, 0.3)) +
    ggplot2::geom_line(data = conf.int, ggplot2::aes(x = x, y = ci.u), lty = "dotted") +
    ggplot2::geom_line(data = conf.int, ggplot2::aes(x = x, y = ci.l), lty = "dotted")


#### Simultaneous confidence bands for a P-P plot
X1 <- rnorm(200)
X2 <- rnorm(200, 1)
x <- seq(0.05, 0.95, length = 19)
y <- EL.smooth("pp", X1, X2, x, conf.level = 0.95,
               simultaneous = TRUE, bw = c(0.3, 0.3))
conf.int <- data.frame(x = x, ci.l = y$conf.int[1,], ci.u = y$conf.int[2,])

## Plot with both pointwise and simultaneous confidence bands
EL.plot("pp", X1, X2, conf.level = 0.95, bw = c(0.3, 0.3)) +
    ggplot2::geom_line(data = conf.int, ggplot2::aes(x = x, y = ci.u), lty = "dotted") +
    ggplot2::geom_line(data = conf.int, ggplot2::aes(x = x, y = ci.l), lty = "dotted")

The two-sample empirical likelihood statistic

Description

Calculates $-2$ times the log-likelihood ratio statistic when the function of interest (either of P-P or Q-Q plot, ROC curve, difference of quantile or distribution functions) at some point t is equal to d.

Usage

EL.statistic(method, X, Y, Delta, d, t, bw = bw.nrd0, conf.level = 0.95)
EL.statistic(method, X, Y, Delta, d, t, bw = bw.nrd0, conf.level = 0.95)

Arguments

method

One of "pp", "qq", "roc", "qdiff", or "fdiff".

X

A numeric vector of data values.

Y

A numeric vector of data values.

Delta

A number. The hypothesised value of the function at point t.

d

Deprecated. Use Delta instead.

t

A number. The point at which to evaluate the statistic.

bw

A function taking a numeric vector and returning a bandwidth, or a numeric vector of length two giving the bandwidths for X and Y respectively. Default is bw.nrd0.

conf.level

Confidence level for the reported confidence interval (default 0.95).

Value

An object of class "htest".

Author(s)

E. Cers, J. Valeinis

References

Valeinis, J. and Cers, E. Extending the two-sample empirical likelihood. Preprint: http://home.lu.lv/~valeinis/lv/petnieciba/EL_TwoSample_2011.pdf

Examples

EL.statistic(method = "pp", X = rnorm(100), Y = rnorm(100), Delta = 0.5, t = 0.5)

EL.statistic(method = "pp", X = rnorm(100), Y = rnorm(100), Delta = 0.5, t = 0.5)

Two-sample frequency-domain empirical likelihood test for autocorrelation difference

Description

Tests whether the lag- $h$ autocorrelation of two independent stationary time series differ by a specified amount $\Delta$ .

Usage

FDEL.acf(
  X,
  Y,
  Delta = 0,
  lag = 1,
  bartlett = FALSE,
  bootstrap.samples = 500,
  center = TRUE,
  seed = NULL,
  span = 0.15,
  rho.lower = -0.99,
  rho.upper = 0.99
)
FDEL.acf(
  X,
  Y,
  Delta = 0,
  lag = 1,
  bartlett = FALSE,
  bootstrap.samples = 500,
  center = TRUE,
  seed = NULL,
  span = 0.15,
  rho.lower = -0.99,
  rho.upper = 0.99
)

Arguments

X, Y

Numeric time-series vectors (length >= 10).

Delta

Hypothesised difference $\rho_x(h) - \rho_y(h) = \Delta$ (default 0).

lag

Positive integer lag $h$ (default 1).

bartlett

Logical; apply a bootstrap Bartlett correction? (default FALSE).

bootstrap.samples

Number of bootstrap replicates for the Bartlett correction (used only when bartlett = TRUE).

center

Logical; subtract sample means before computing periodograms? (default TRUE).

seed

Optional integer seed for reproducibility.

span

Loess span for periodogram smoothing used in the Bartlett correction (default 0.15).

rho.lower, rho.upper

Search bounds for the profile autocorrelation optimization (defaults -0.99, 0.99).

Value

An object of class c("FDELacf", "htest") with components statistic, parameter, p.value, estimate, null.value, alternative, method, data.name, lag, bartlett, bartlett.factor, statistic.uncorrected, p.value.uncorrected, B, call.

Author(s)

R. Alksnis, J. Valeinis

Examples

set.seed(1)
X <- arima.sim(n = 200, model = list(ar = 0.3))
Y <- arima.sim(n = 350, model = list(ar = 0.3))
FDEL.acf(X, Y)
## With Bartlett correction (slower)
## FDEL.acf(X, Y, bartlett = TRUE, B = 199)

set.seed(1)
X <- arima.sim(n = 200, model = list(ar = 0.3))
Y <- arima.sim(n = 350, model = list(ar = 0.3))
FDEL.acf(X, Y)
## With Bartlett correction (slower)
## FDEL.acf(X, Y, bartlett = TRUE, B = 199)

Package 'EL'

Help Index

The two-sample blockwise empirical likelihood statistic for differences in means

Description

Usage

Arguments

Value

Author(s)

Examples

Empirical likelihood test for the difference of smoothed Huber estimators

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Empirical likelihood test for the difference of two sample means

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Draws plots using the smoothed two-sample empirical likelihood method

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Smooth estimates and confidence intervals using the smoothed two-sample EL method

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The two-sample empirical likelihood statistic

Description

Usage

Arguments

Value

Author(s)

References

See Also

Examples

Two-sample frequency-domain empirical likelihood test for autocorrelation difference

Description

Usage

Arguments

Value

Author(s)

Examples