# 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)`

```
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, ")"))
}
p <- c(1, 1, 1, 2, 2:6, 8, 10, 11, 13, 15, 17, 20, 22, 24,
27, 29, 31, 33, 35, 36, 38, 39, 39, 40)
stopifnot(round(dsignrank(0:56, n = 10)* 2^10) == c(p, rev(p), 0),
qsignrank((1:16)/ 16, n = 4) == c(0:2, rep(3:7, each = 2), 8:10))
```

*Documentation reproduced from package stats, version 3.3, License: Part of R 3.3*