Non-parametric tests for the two-sample problem based on order statistics and power comparisons

tnl.test performs a nonparametric test for two sample test on vectors of data.

ptnl gives the distribution function of \(T_n^{(\ell)}\) against the specified quantiles.

dtnl gives the density of \(T_n^{(\ell)}\) against the specified quantiles.

qtnl gives the quantile function of \(T_n^{(\ell)}\) against the specified probabilities.

rtnl generates random values from \(T_n^{(\ell)}\).

tnl_mean() gives an expression for \(E(T_n^{(\ell)})\) under \(H_0:F=G\).

ptnl.lehmann gives the distribution function of \(T_n^{(\ell)}\) under Lehmann alternatives.

dtnl.lehmann gives the density of \(T_n^{(\ell)}\) under Lehmann alternatives.

qtnl.lehmann gives the quantile function of \(T_n^{(\ell)}\) against the specified probabilities under Lehmann alternatives.

rtnl.lehmann generates random values from \(T_n^{(\ell)}\) under Lehmann alternatives.

Usage

tnl.test(x, y, l, exact = "NULL")

ptnl(q, n, m, l, exact = "NULL", trial = 1e+05)

dtnl(k, n, m, l, exact = "NULL", trial = 1e+05)

qtnl(p, n, m, l, exact = "NULL", trial = 1e+05)

rtnl(N, n, m, l)

tnl_mean(n., m., l)

ptnl.lehmann(q, n., m., l, gamma)

dtnl.lehmann(k, n., m., l, gamma)

qtnl.lehmann(p, n., m., l, gamma)

rtnl.lehmann(N, n., m., l, gamma)

Arguments

x

the first (non-empty) numeric vector of data values.

y

the second (non-empty) numeric vector of data values.

l

class parameter of \(T_n^{(\ell)}\).

exact

the method that will be used. "NULL" or a logical indicating whether an exact should be computed. See 'Details' for the meaning of NULL.

n, m

samples size.

trial

number of trials for simulation.

k, q

vector of quantiles.

p

vector of probabilities.

N

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

n., m.

samples size.

gamma

parameter of Lehmann alternative.

Value

tnl.test returns a list with the following components

statistic:

the value of the test statistic.

p.value:

the p-value of the test.

ptnl returns a list with the following components

method:

The method that was used (exact or simulation). See 'Details'.

cdf:

distribution function of \(T_n^{(\ell)}\) against the specified quantiles.

dtnl returns a list with the following components

method:

The method that was used (exact or simulation). See 'Details'.

pmf:

density of \(T_n^{(\ell)}\) against the specified quantiles.

qtnl returns a list with the following components

method:

The method that was used (exact or simulation). See 'Details'.

quantile:

quantile function against the specified probabilities.

rtnl return N of the generated random values.

tnl_mean() return the mean of \(T_n^{(\ell)}\).

ptnl.lehmann return vector of the distribution under Lehmann alternatives against the specified gamma.

dtnl.lehmann return vector of the density under Lehmann alternatives against the specified gamma.

qtnl.lehmann returns a quantile function against the specified probabilities under Lehmann alternatives.

rtnl.lehmann return N of the generated random values under Lehmann alternatives.

Details

A non-parametric two-sample test is performed for testing null hypothesis \(H_0:F=G\) against the alternative hypothesis \(H_1:F\not= G\). The assumptions of the \(T_n^{(\ell)}\) test are that both samples should come from a continuous distribution and the samples should have the same sample size.

Missing values are silently omitted from \(x\) and \(y\).

Exact and simulated p-values are available for the \(T_n^{(\ell)}\) test. If exact ="NULL" (the default) the p-value is computed based on exact distribution when the sample size is less than 11. Otherwise, p-value is computed based on a Monte Carlo simulation. If exact ="TRUE", an exact p-value is computed. If exact="FALSE" , a Monte Carlo simulation is performed to compute the p-value. It is recommended to calculate the p-value by a Monte Carlo simulation (use exact="FALSE"), as it takes too long to calculate the exact p-value when the sample size is greater than 10.

The probability mass function (pmf), cumulative density function (cdf) and quantile function of \(T_n^{(\ell)}\) are also available in this package, and the above-mentioned conditions about exact ="NULL", exact ="TRUE" and exact="FALSE" is also valid for these functions.

Exact distribution of \(T_n^{(\ell)}\) test is also computed under Lehman alternative.

Random number generator of \(T_n^{(\ell)}\) test statistic are provided under null hypothesis in the library.

References

Karakaya, K., Sert, S., Abusaif, I., Kuş, C., Ng, H. K. T., & Nagaraja, H. N. (2023). A Class of Non-parametric Tests for the Two-Sample Problem based on Order Statistics and Power Comparisons. Submitted paper.

Aliev, F., Özbek, L., Kaya, M. F., Kuş, C., Ng, H. K. T., & Nagaraja, H. N. (2022). A nonparametric test for the two-sample problem based on order statistics. Communications in Statistics-Theory and Methods, 1-25.

Examples

#require(stats)
#x <- rnorm(7, 2, 0.5)
#y <- rnorm(5, 0, 1)
#tnl.test(x, y, l = 2)
## $statistic
## [1] 2
##
## $p.value
## [1]  0.01515152
#ptnl(q = c(2, 5), n = 6, m = 5, l = 2, trial = 100000)
## $method
## [1] "exact"
##
## $cdf
## [1] 0.02164502 1.00000000
#dtnl(k = c(1, 3, 6), n = 7, m = 5, l = 2)
## $method
## [1] "exact"
##
## $pmf
## [1] 0.00000000 0.04671717 0.00000000
#qtnl(p = c(.3, .9), n = 4, m = 5, l = 1)
## $method
## [1] "exact"
##
## $quantile
## [1]  3 4
#rtnl(N = 20, n = 7, m = 10, l = 1)
## [1] 7 6 7 4 6 6 6 5 5 6 6 3 2 4 3 7 7 7 7 6
#require(base)
#tnl_mean(n. = 11, m. = 8, l = 1)
## [1] 5.1693
#ptnl.lehmann(q = 3, n. = 5, m. = 7, l = 2, gamma = 1.2)
## [1] 0.07471397
#dtnl.lehmann(k = 3, n. = 6, m. = 5, l = 2, gamma = 0.8)
## [1] 0.07073467
#qtnl.lehmann(p = c(.1, .5, .9), n. = 7, m. = 5, l = 1, gamma = 0.5)
## [1] 1 3 5
# rtnl.lehmann(N = 15, n = 7,m=7, l = 2, gamma = 0.5)
## [1] 4 4 7 5 3 7 7 4 4 5 5 7 7 3