Title:	'R6'-Based Flexible Framework for Permutation Tests
Version:	1.2.9
Description:	Implements non-parametric tests from Higgins (2004, ISBN:0534387756), including tests for one sample, two samples, k samples, paired comparisons, blocked designs, trends and association. Built with 'Rcpp' for efficiency and 'R6' for flexible, object-oriented design, the package provides a unified framework for performing or creating custom permutation tests.
BugReports:	https://github.com/qddyy/LearnNonparam/issues
URL:	https://github.com/qddyy/LearnNonparam, https://qddyy.github.io/LearnNonparam/
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Depends:	R (≥ 3.5.0)
LinkingTo:	Rcpp (≥ 1.0.10)
Imports:	R6 (≥ 2.5.0), Rcpp (≥ 1.0.10), stats, graphics, compiler
Suggests:	ggplot2
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.3.2
NeedsCompilation:	yes
Packaged:	2025-06-02 07:45:56 UTC; dy
Author:	Yan Du [aut, cre]
Maintainer:	Yan Du <isduyan@outlook.com>
Repository:	CRAN
Date/Publication:	2025-06-02 08:10:02 UTC

Ansari-Bradley Test

Description

Performs Ansari-Bradley test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> AnsariBradley

Methods

Public methods

• AnsariBradley$new()

Method new()

Create a new AnsariBradley object.

Usage

AnsariBradley$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000
)

Arguments

Returns

A AnsariBradley object.

Examples

pmt(
    "twosample.ansari",
    alternative = "greater", n_permu = 0
)$test(Table2.8.1)$print()

Inference on Cumulative Distribution Function

Description

Performs statistical inference on population cumulative distribution function.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::OneSampleTest -> CDF

Methods

Public methods

• CDF$new()

• CDF$plot()

Method new()

Create a new CDF object.

Usage

CDF$new(method = c("binomial", "dkw"), conf_level = 0.95)

Arguments

Returns

A CDF object.

Method plot()

Plot the estimate and confidence bounds for population cumulative distribution function.

Usage

CDF$plot(style = c("graphics", "ggplot2"))

Arguments

Returns

The object itself (invisibly).

Examples

pmt("onesample.cdf")$test(Table1.2.1)$plot(style = "graphic")

Chi-Square Test on Contingency Table

Description

Performs chi-square test on contingency tables.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::ContingencyTableTest -> ChiSquare

Methods

Public methods

• ChiSquare$new()

Method new()

Create a new ChiSquare object.

Usage

ChiSquare$new(type = c("permu", "asymp"), n_permu = 10000)

Arguments

Returns

A ChiSquare object.

Examples

t <- pmt(
    "table.chisq", n_permu = 0
)$test(Table5.4.2)$print()

t$type <- "asymp"
t

ContingencyTableTest Class

Description

Abstract class for tests on contingency tables.

Super class

LearnNonparam::PermuTest -> ContingencyTableTest

Test for Association Between Paired Samples

Description

Performs correlation coefficient based two-sample association test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSamplePairedTest -> LearnNonparam::TwoSampleAssociationTest -> Correlation

Methods

Public methods

• Correlation$new()

Method new()

Create a new Correlation object.

Usage

Correlation$new(
  type = c("permu", "asymp"),
  method = c("pearson", "kendall", "spearman"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000
)

Arguments

Returns

A Correlation object.

Examples

pmt(
    "association.corr", method = "pearson",
    alternative = "greater", n_permu = 10000
)$test(Table5.1.2)$print()

t <- pmt(
    "association.corr", method = "spearman",
    alternative = "two_sided", n_permu = 10000
)$test(Table5.1.2)$print()

t$type <- "asymp"
t

t <- pmt(
    "association.corr", method = "kendall",
    alternative = "greater", n_permu = 0
)$test(Table5.2.2)$print()

t$type <- "asymp"
t

Two-Sample Test Based on Mean or Median

Description

Performs mean/median based two-sample test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSampleLocationTest -> Difference

Methods

Public methods

• Difference$new()

Method new()

Create a new Difference object.

Usage

Difference$new(
  method = c("mean", "median"),
  alternative = c("two_sided", "less", "greater"),
  null_value = 0,
  n_permu = 10000
)

Arguments

Returns

A Difference object.

Examples

pmt(
    "twosample.difference", method = "mean",
    alternative = "greater", n_permu = 0
)$test(Table2.1.1)$print()$plot(
    style = "graphic", breaks = seq(-20, 25, length.out = 9)
)

pmt(
    "twosample.difference", method = "mean",
    alternative = "greater", n_permu = 1000
)$test(Table2.3.1)$print()

Friedman Test

Description

Performs Friedman test on samples collected in a randomized complete block design.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::RCBDTest -> Friedman

Methods

Public methods

• Friedman$new()

Method new()

Create a new Friedman object.

Usage

Friedman$new(type = c("permu", "asymp"), n_permu = 10000)

Arguments

Returns

A Friedman object.

Examples

t <- pmt(
    "rcbd.friedman", n_permu = 0
)$test(Table4.5.3)$print()

t$type <- "asymp"
t

Jonckheere-Terpstra Test

Description

Performs Jonckheere-Terpstra test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::KSampleTest -> JonckheereTerpstra

Methods

Public methods

• JonckheereTerpstra$new()

Method new()

Create a new JonckheereTerpstra object.

Usage

JonckheereTerpstra$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000
)

Arguments

Returns

A JonckheereTerpstra object.

Examples

t <- pmt(
    "ksample.jt", alternative = "greater"
)$test(Table3.4.1)$print()

t$type <- "asymp"
t

KSampleTest Class

Description

Abstract class for k-sample tests.

Super class

LearnNonparam::PermuTest -> KSampleTest

Two-Sample Kolmogorov-Smirnov Test

Description

Performs two-sample Kolmogorov-Smirnov test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> KolmogorovSmirnov

Methods

Public methods

• KolmogorovSmirnov$new()

Method new()

Create a new KolmogorovSmirnov object.

Usage

KolmogorovSmirnov$new(n_permu = 10000)

Arguments

Returns

A KolmogorovSmirnov object.

Examples

pmt(
    "twosample.ks", n_permu = 0
)$test(Table2.8.1)$print()

Kruskal-Wallis Test

Description

Performs Kruskal-Wallis test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::KSampleTest -> KruskalWallis

Methods

Public methods

• KruskalWallis$new()

Method new()

Create a new KruskalWallis object.

Usage

KruskalWallis$new(
  type = c("permu", "asymp"),
  scoring = c("rank", "vw", "expon"),
  n_permu = 10000
)

Arguments

Returns

A KruskalWallis object.

Examples

pmt(
    "ksample.kw", type = "asymp"
)$test(Table3.2.2)$print()

t <- pmt(
    "ksample.kw", type = "permu"
)$test(Table3.2.3)$print()

t$type <- "asymp"
t

MultipleComparison Class

Description

Abstract class for multiple comparisons.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::KSampleTest -> MultipleComparison

OneSampleTest Class

Description

Abstract class for one-sample tests.

Super class

LearnNonparam::PermuTest -> OneSampleTest

Methods

Public methods

• OneSampleTest$plot()

Method plot()

Usage

OneSampleTest$plot(...)

Arguments

One-Way Test for Equal Means

Description

Performs F statistic based one-way test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::KSampleTest -> OneWay

Methods

Public methods

• OneWay$new()

Method new()

Create a new OneWay object.

Usage

OneWay$new(type = c("permu", "asymp"), n_permu = 10000)

Arguments

Returns

A OneWay object.

Examples

t <- pmt(
    "ksample.oneway", n_permu = 0
)$test(Table3.1.2)$print()

t$type <- "asymp"
t

Page Test

Description

Performs Page test on samples collected in a randomized complete block design.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::RCBDTest -> Page

Methods

Public methods

• Page$new()

Method new()

Create a new Page object.

Usage

Page$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000
)

Arguments

Returns

A Page object.

Examples

t <- pmt(
    "rcbd.page", alternative = "less"
)$test(Table4.4.3)$print()

t$type <- "asymp"
t

Paired Comparison Based on Differences

Description

Performs differences based paired comparison on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSamplePairedTest -> PairedDifference

Active bindings

Methods

Public methods

• PairedDifference$new()

Method new()

Create a new PairedDifference object.

Usage

PairedDifference$new(
  type = c("permu", "asymp"),
  method = c("with_zeros", "without_zeros"),
  scoring = c("none", "rank", "vw", "expon"),
  alternative = c("two_sided", "less", "greater"),
  null_value = 0,
  n_permu = 10000,
  correct = TRUE
)

Arguments

Returns

A PairedDifference object.

Examples

pmt(
    "paired.difference",
    alternative = "greater", scoring = "none", n_permu = 0
)$test(Table4.1.1)$print()

pmt(
    "paired.difference", n_permu = 0
)$test(Table4.1.3)$print()

t <- pmt(
    "paired.difference", scoring = "rank",
    alternative = "greater", n_permu = 0
)$test(Table4.1.1)$print()

t$type <- "asymp"
t

PermuTest Class

Description

Abstract class for permutation tests.

Active bindings

Methods

Public methods

• PermuTest$test()

• PermuTest$print()

• PermuTest$plot()

Method test()

Perform test on sample(s).

Usage

PermuTest$test(...)

Arguments

Details

A progress bar is shown by default. Use options(LearnNonparam.pmt_progress = FALSE) to disable it.

Returns

The object itself (invisibly).

Method print()

Print the results of the test.

Usage

PermuTest$print()

Returns

The object itself (invisibly).

Method plot()

Plot histogram(s) of the permutation distribution. Note that this method only works if type is set to "permu".

Usage

PermuTest$plot(style = c("graphics", "ggplot2"), ...)

Arguments

Returns

The object itself (invisibly).

Quantile Test

Description

Performs quantile test on a single sample. In addition, an estimation and a confidence interval for the desired quantile will be calculated.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::OneSampleTest -> Quantile

Active bindings

Methods

Public methods

• Quantile$new()

Method new()

Create a new Quantile object.

Usage

Quantile$new(
  type = c("asymp", "exact"),
  alternative = c("two_sided", "less", "greater"),
  null_value = 0,
  conf_level = 0.95,
  prob = 0.5,
  correct = TRUE
)

Arguments

Returns

A Quantile object.

Examples

pmt(
    "onesample.quantile", prob = 0.5,
    null_value = 75, alternative = "greater",
    type = "asymp", correct = FALSE
)$test(Table1.1.1)$print()

pmt(
    "onesample.quantile",
    prob = 0.25, conf_level = 0.90
)$test(Table1.2.1)$conf_int

One-Way Test for Equal Means in RCBD

Description

Performs F statistic based one-way test on samples collected in a randomized complete block design.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::RCBDTest -> RCBDOneWay

Methods

Public methods

• RCBDOneWay$new()

Method new()

Create a new RCBDOneWay object.

Usage

RCBDOneWay$new(type = c("permu", "asymp"), n_permu = 10000)

Arguments

Returns

A RCBDOneWay object.

Examples

t <- pmt(
    "rcbd.oneway", n_permu = 5000
)$test(Table4.4.3)$print()

t$type <- "asymp"
t

RCBDTest Class

Description

Abstract class for tests on samples collected in randomized complete block designs.

Super class

LearnNonparam::PermuTest -> RCBDTest

Ratio Mean Deviance Test

Description

Performs ratio mean deviance test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> RatioMeanDeviance

Methods

Public methods

• RatioMeanDeviance$new()

Method new()

Create a new RatioMeanDeviance object.

Usage

RatioMeanDeviance$new(
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000
)

Arguments

Returns

A RatioMeanDeviance object.

Examples

pmt(
    "twosample.rmd",
    alternative = "greater", n_permu = 0
)$test(Table2.8.1)$print()

Two-Sample Test Based on Sum of Scores

Description

Performs sum of scores based two-sample test on samples. It is almost the same as two-sample wilcoxon rank sum test but uses more scoring systems.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSampleLocationTest -> ScoreSum

Methods

Public methods

• ScoreSum$new()

Method new()

Create a new ScoreSum object.

Usage

ScoreSum$new(
  scoring = c("rank", "vw", "expon"),
  alternative = c("two_sided", "less", "greater"),
  null_value = 0,
  n_permu = 10000
)

Arguments

Returns

A ScoreSum object.

Examples

pmt(
    "twosample.scoresum", scoring = "expon",
    alternative = "greater", n_permu = 0
)$test(Table2.6.2)$print()

Siegel-Tukey Test

Description

Performs Siegel-Tukey test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSampleLocationTest -> LearnNonparam::Wilcoxon -> SiegelTukey

Methods

Public methods

• SiegelTukey$new()

Method new()

Create a new SiegelTukey object.

Usage

SiegelTukey$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000,
  correct = TRUE
)

Arguments

Returns

A SiegelTukey object.

Examples

pmt(
    "twosample.siegel",
    alternative = "greater", n_permu = 0
)$test(Table2.8.1)$print()

Two-Sample Sign Test

Description

Performs two-sample sign test on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSamplePairedTest -> Sign

Active bindings

Methods

Public methods

• Sign$new()

Method new()

Create a new Sign object.

Usage

Sign$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  n_permu = 10000,
  correct = TRUE
)

Arguments

Returns

A Sign object.

Examples

t <- pmt(
    "paired.sign",
    alternative = "greater", n_permu = 0
)$test(
    rep(c(+1, -1), c(12, 5)), rep(0, 17)
)$print()

t$type <- "asymp"
t

Multiple Comparison Based on Studentized Statistic

Description

Performs studentized statistic based multiple comparison on samples.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::KSampleTest -> LearnNonparam::MultipleComparison -> Studentized

Methods

Public methods

• Studentized$new()

Method new()

Create a new Studentized object.

Usage

Studentized$new(
  type = c("permu", "asymp"),
  method = c("bonferroni", "tukey"),
  scoring = c("none", "rank", "vw", "expon"),
  conf_level = 0.95,
  n_permu = 10000
)

Arguments

Returns

A Studentized object.

Examples

t <- pmt(
    "multcomp.studentized", method = "bonferroni"
)$test(Table3.3.1)$print()

t$type <- "asymp"
t

t$scoring <- "rank"
t

t$method <- "tukey"
t

t$scoring <- "none"
t

t$type <- "permu"
t

Sodium Contents

Description

Sodium contents (in mg) of 40 servings of a food product.

Usage

Table1.1.1

Format

An object of class numeric of length 40.

Source

Table 1.1.1

Cycles Until Failure

Description

The number of cycles (in thousands) that it takes for 20 door latches to fail.

Usage

Table1.2.1

Format

An object of class numeric of length 20.

Source

Table 1.2.1

Test Scores

Description

Test scores of 7 employees for comparison of methods of instruction.

Usage

Table2.1.1

Format

An object of class list of length 2.

Source

Table 2.1.1

Runoff Minutes

Description

The numbers of minutes it took to obtain various amounts of runoff on each plot.

Usage

Table2.3.1

Format

An object of class data.frame with 8 rows and 2 columns.

Source

Table 2.3.1

Hours Until Recharge

Description

The numbers of hours that 2 brands of laptop computers function before battery recharging is necessary.

Usage

Table2.6.1

Format

An object of class data.frame with 4 rows and 2 columns.

Source

Table 2.6.1

Cerium Amounts

Description

The amounts of cerium measured in samples of granite and basalt.

Usage

Table2.6.2

Format

An object of class data.frame with 6 rows and 2 columns.

Source

Table 2.6.2

Ounces Of Beverage

Description

The amounts of liquid in randomly selected beverage containers before and after the filling process has been repaired.

Usage

Table2.8.1

Format

An object of class data.frame with 5 rows and 2 columns.

Source

Table 2.8.1

Normal Samples

Description

Observations randomly sampled from normal populations with means 15, 25 and 30, respectively, and standard deviation 9.

Usage

Table3.1.2

Format

An object of class data.frame with 5 rows and 3 columns.

Source

Table 3.1.2

Logarithms of Bacteria Counts

Description

Logarithms of counts of bacteria in 4 samples, which respectively were treated with 3 kills and left untreated for the control.

Usage

Table3.2.2

Format

An object of class list of length 4.

Source

Table 3.2.2

Saltiness Scores

Description

Saltiness scores, on a scale of 1 to 5, assigned by a taste expert to samples of 3 food products that differ in the amounts of soymeal they contain.

Usage

Table3.2.3

Format

An object of class list of length 3.

Source

Table 3.2.3

Percentages of Clay

Description

The percentages of clay in 6 samples of soil selected from 4 locations.

Usage

Table3.3.1

Format

An object of class data.frame with 6 rows and 4 columns.

Source

Table 3.3.1

Phosphorus Contents

Description

Phosphorus contents of plants under 4 mowing treatments.

Usage

Table3.4.1

Format

An object of class data.frame with 6 rows and 4 columns.

Source

Table 3.4.1

Caloric Intake

Description

The estimated daily caloric intake from dietary information provided using 2 methods by a group of college women.

Usage

Table4.1.1

Format

An object of class data.frame with 5 rows and 2 columns.

Source

Table 4.1.1

Cholesterol Reduction

Description

Reduction in cholesterol after twins were given 2 drugs separately.

Usage

Table4.1.3

Format

An object of class data.frame with 17 rows and 2 columns.

Source

Table 4.1.3

Yield Data

Description

Yield data for a randomized complete block design in which 4 different types of tractors were used in tilling the soil. The blocking factor is location of the fields.

Usage

Table4.4.3

Format

An object of class data.frame with 4 rows and 6 columns.

Source

Table 4.4.3

Randomized Complete Block with Ties

Description

A randomized complete block design with 4 treatments and 3 blocks.

Usage

Table4.5.3

Format

An object of class data.frame with 4 rows and 3 columns.

Source

Table 4.5.3

Heterophils and Lymphocytes

Description

Counts of the heterophils and lymphocytes in blood samples from 18 healthy rabbits.

Usage

Table5.1.2

Format

An object of class data.frame with 18 rows and 2 columns.

Source

Table5.1.2

Scores of Projects

Description

Scores of 10 projects at a science fair.

Usage

Table5.2.2

Format

An object of class data.frame with 10 rows and 2 columns.

Source

Table5.2.2

Satisfaction with Pain-Relief Treatment

Description

Patients' responses with 2 methods of relieving postoperative pain.

Usage

Table5.4.2

Format

An object of class data.frame with 2 rows and 3 columns.

Source

Table5.4.2

TwoSampleAssociationTest Class

Description

Abstract class for two-sample association tests.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSamplePairedTest -> TwoSampleAssociationTest

TwoSampleLocationTest Class

Description

Abstract class for two-sample location tests.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> TwoSampleLocationTest

TwoSamplePairedTest Class

Description

Abstract class for paired two-sample tests.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> TwoSamplePairedTest

TwoSampleTest Class

Description

Abstract class for two-sample tests.

Super class

LearnNonparam::PermuTest -> TwoSampleTest

Two-Sample Wilcoxon Test

Description

Performs two-sample wilcoxon test on samples. In addition, an estimation and a confidence interval for the location shift will be calculated.

Super classes

LearnNonparam::PermuTest -> LearnNonparam::TwoSampleTest -> LearnNonparam::TwoSampleLocationTest -> Wilcoxon

Active bindings

Methods

Public methods

• Wilcoxon$new()

Method new()

Create a new Wilcoxon object.

Usage

Wilcoxon$new(
  type = c("permu", "asymp"),
  alternative = c("two_sided", "less", "greater"),
  null_value = 0,
  conf_level = 0.95,
  n_permu = 10000,
  correct = TRUE
)

Arguments

Returns

A Wilcoxon object.

Examples

pmt(
    "twosample.wilcoxon",
    alternative = "greater", n_permu = 0
)$test(Table2.1.1)$print()

pmt(
    "twosample.wilcoxon",
    alternative = "less", n_permu = 0
)$test(Table2.6.1)$print()

pmt(
    "twosample.wilcoxon", conf_level = 0.90
)$test(Table2.6.2)$conf_int

Syntactic Sugar for Object Construction

Description

Construct test objects in a unified way.

Usage

pmt(key, ...)

pmts(
  which = c("all", "onesample", "twosample", "ksample", "multcomp", "paired", "rcbd",
    "association", "table")
)

define_pmt(
  inherit = c("twosample", "ksample", "paired", "rcbd", "association", "table"),
  statistic,
  rejection = c("lr", "l", "r"),
  scoring = "none",
  n_permu = 10000,
  name = "User-Defined Permutation Test",
  alternative = NULL,
  depends = character(),
  plugins = character(),
  includes = character()
)

Arguments

Details

The test statistic can be defined using either R or Rcpp, with the statistic parameter specified as:

• R: a function returning a closure that returns a double.

• Rcpp: a character string defining a captureless lambda (since C++11) returning another lambda that captures by value, accepts parameters of the same type, and returns a double.

The purpose of this design is to pre-calculate certain constants that remain invariant during permutation.

When using Rcpp, the parameters for different inherit are listed as follows. Note that the names can be customized, and the types can be replaced with auto (thanks to the support for generic lambdas in C++14). See examples.

When using R, the parameters should be the R equivalents of these.

Value

a test object corresponding to the specified key.

a data frame containing keys and corresponding tests implemented in this package.

a test object based on the specified statistic.

Note

• statistic should not cause errors or return missing values.

• The data is permuted in-place. Therefore, modifications to the data within statistic may lead to incorrect results. Since R has copy-on-modify semantics but C++ does not, it is recommended to pass const references when using Rcpp in define_pmt, as shown in the table above.

Examples

pmt("twosample.wilcoxon")

pmts("ksample")

x <- rnorm(5)
y <- rnorm(5, 1)

t <- define_pmt(
    inherit = "twosample",
    scoring = base::rank, # equivalent to "rank"
    statistic = function(...) function(x, y) sum(x)
)$test(x, y)$print()

t$scoring <- function(x) qnorm(rank(x) / (length(x) + 1)) # equivalent to "vw"
t$print()

t$n_permu <- 0
t$print()


r <- define_pmt(
    inherit = "twosample", n_permu = 1e5,
    statistic = function(x, y) {
        m <- length(x)
        n <- length(y)
        function(x, y) sum(x) / m - sum(y) / n
    }
)


rcpp <- define_pmt(
    inherit = "twosample", n_permu = 1e5,
    statistic = "[](const auto& x, const auto& y) {
        auto m = x.length();
        auto n = y.length();
        return [=](const auto& x, const auto& y) {
            return sum(x) / m - sum(y) / n;
        };
    }"
)

# equivalent
# rcpp <- define_pmt(
#     inherit = "twosample", n_permu = 1e5,
#     statistic = "[](const NumericVector& x, const NumericVector& y) {
#         R_xlen_t m = x.length();
#         R_xlen_t n = y.length();
#         return [m, n](const NumericVector& x, const NumericVector& y) -> double {
#             return sum(x) / m - sum(y) / n;
#         };
#     }"
# )

options(LearnNonparam.pmt_progress = FALSE)
system.time(r$test(x, y))
system.time(rcpp$test(x, y))