The goal of tnl.Test is to provide functions to perform the hypothesis tests for the two sample problem based on order statistics and power comparisons.

You can install the released version of tnl.Test from CRAN with:

Alternatively, you can install the development version on GitHub using the devtools package:

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.

`tnl.test`

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

```
library(tnl.Test)
require(stats)
x=rnorm(7,2,0.5)
y=rnorm(7,0,1)
tnl.test(x,y,l=2)
#> $statistic
#> [1] 4
#>
#> $p.value
#> [1] 0.1818182
```

`ptnl`

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

```
library(tnl.Test)
ptnl(q=2,n=6,m=9,l=2,exact="NULL")
#> $method
#> [1] "exact"
#>
#> $cdf
#> [1] 0.01198801
```

`dtnl`

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

```
library(tnl.Test)
dtnl(k=3,n=7,m=10,l=2,exact="TRUE")
#> $method
#> [1] "exact"
#>
#> $pmf
#> [1] 0.02303579
```

`qtnl`

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

```
library(tnl.Test)
qtnl(p=c(.1,.3,.5,.8,1),n=8,m=8,l=1,exact="NULL",trial = 100000)
#> $method
#> [1] "exact"
#>
#> $quantile
#> [1] 2 3 4 6 8
```

`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`

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

`rtnl.lehmann`

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

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.