# SignRank

##### Distribution of the Wilcoxon Signed Rank Statistic

Density, distribution function, quantile function and random
generation for the distribution of the Wilcoxon Signed Rank statistic
obtained from a sample with size `n`

.

- Keywords
- distribution

##### Usage

```
dsignrank(x, n, log = FALSE)
psignrank(q, n, lower.tail = TRUE, log.p = FALSE)
qsignrank(p, n, lower.tail = TRUE, log.p = FALSE)
rsignrank(nn, n)
```

##### Arguments

- x, q
vector of quantiles.

- p
vector of probabilities.

- nn
number of observations. If

`length(nn) > 1`

, the length is taken to be the number required.- n
number(s) of observations in the sample(s). A positive integer, or a vector of such integers.

- log, log.p
logical; if TRUE, probabilities p are given as log(p).

- lower.tail
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

##### Details

This distribution is obtained as follows. Let `x`

be a sample of
size `n`

from a continuous distribution symmetric about the
origin. Then the Wilcoxon signed rank statistic is the sum of the
ranks of the absolute values `x[i]`

for which `x[i]`

is
positive. This statistic takes values between \(0\) and
\(n(n+1)/2\), and its mean and variance are \(n(n+1)/4\) and
\(n(n+1)(2n+1)/24\), respectively.

If either of the first two arguments is a vector, the recycling rule is used to do the calculations for all combinations of the two up to the length of the longer vector.

##### Value

`dsignrank`

gives the density,
`psignrank`

gives the distribution function,
`qsignrank`

gives the quantile function, and
`rsignrank`

generates random deviates.

The length of the result is determined by `nn`

for
`rsignrank`

, and is the maximum of the lengths of the
numerical arguments for the other functions.

The numerical arguments other than `nn`

are recycled to the
length of the result. Only the first elements of the logical
arguments are used.

##### See Also

`wilcox.test`

to calculate the statistic from data, find p
values and so on.

Distributions for standard distributions, including
`dwilcox`

for the distribution of *two-sample*
Wilcoxon rank sum statistic.

##### Examples

`library(stats)`

```
# NOT RUN {
require(graphics)
par(mfrow = c(2,2))
for(n in c(4:5,10,40)) {
x <- seq(0, n*(n+1)/2, length = 501)
plot(x, dsignrank(x, n = n), type = "l",
main = paste0("dsignrank(x, n = ", n, ")"))
}
# }
```

*Documentation reproduced from package stats, version 3.6.0, License: Part of R 3.6.0*