---
title: "tnl_Test"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{tnl_Test}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r setup}
library(tnl.Test)
```
 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.


## Installation

You can install the released version of tnl.Test from [CRAN](https://CRAN.R-project.org) with:

``` r
install.packages("tnl.Test")
```
Alternatively, you can install the development version on GitHub using the devtools package:
``` r
install.packages("devtools") # if you have not installed "devtools" package
devtools::install_github("ihababusaif/tnl.Test")
```
## 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.<br />
Missing values are silently omitted from x and y.<br />
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. <br />
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.<br />
Exact distribution of ${T_n^{(\ell)}}$ test is also computed under Lehman alternative.<br />
Random number generator of ${T_n^{(\ell)}}$ test statistic are provided under null hypothesis in the library.


## Examples

```tnl.test``` function performs a nonparametric test for two sample test on vectors of data.
```{r}
library(tnl.Test)
require(stats)
 x=rnorm(7,2,0.5)
 y=rnorm(7,0,1)
 tnl.test(x,y,l=2)
```

```ptnl``` gives the distribution function of ${T_n^{(\ell)}}$ against the specified quantiles.
```{r}
library(tnl.Test)
 ptnl(q=2,n=6,m=9,l=2,exact="NULL")
```

```dtnl``` gives the density of ${T_n^{(\ell)}}$ against the specified quantiles.
```{r}
library(tnl.Test)
 dtnl(k=3,n=7,m=10,l=2,exact="TRUE")
```

```qtnl``` gives the quantile function of ${T_n^{(\ell)}}$ against the specified probabilities.
```{r}
library(tnl.Test)
 qtnl(p=c(.1,.3,.5,.8,1),n=8,m=8,l=1,exact="NULL",trial = 100000)
```

```rtnl``` generates random values from ${T_n^{(\ell)}}$.
```{r}
library(tnl.Test)
 rtnl(N=15,n=7,m=10,l=2)
```

```tnl_mean``` gives an expression for $E({T_n^{(\ell)}})$ under ${H_0:F=G}$.
```{r}
library(tnl.Test)
require(base)
 tnl_mean(n.=11,m.=8, l=2)
```


```ptnl.lehmann``` gives the  distribution function of ${T_n^{(\ell)}}$ under Lehmann alternatives.
```{r}
library(tnl.Test)
ptnl.lehmann(q=3, n.=7,m.=7,l = 2, gamma = 1.2)
```


```dtnl.lehmann``` gives the density of ${T_n^{(\ell)}}$ under Lehmann alternatives. 
```{r}
library(tnl.Test)
 dtnl.lehmann(k=3, n.= 6,m.=8,l = 2, gamma = 0.8)
```


```qtnl.lehmann``` returns a quantile function against the specified probabilities under Lehmann alternatives.                      
```{r}
library(tnl.Test)
qtnl.lehmann(p=.3, n.=4,m.=7, l=1, gamma=0.5)
```

```rtnl.lehmann``` generates random values from ${T_n^{(\ell)}}$ under Lehmann alternatives.
```{r}
library(tnl.Test)
rtnl.lehmann(N = 15, n. = 7,m.=10, l = 2,gamma=0.5)
```

## Corresponding Author
Department of Statistics, Faculty of Science, Selcuk University, 42250, Konya, Turkey  <br />
www.researchgate.net/profile/Ihab-Abusaif  <br />
Email:censtat@gmail.com


## 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.<br />

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.