# runs: Distribution of the Wald Wolfowitz Runs Statistic In randtests: Testing randomness in R

## Description

Probability function, distribution function, quantile function and random generation for the distribution of the Runs statistic obtained from samples with n1 and n2 elements of each type.

## Usage

 ```1 2 3 4``` ```druns(x, n1, n2, log = FALSE) pruns(q, n1, n2, lower.tail = TRUE, log.p = FALSE) qruns(p, n1, n2, lower.tail = TRUE, log.p = FALSE) rruns(n, n1, n2) ```

## Arguments

 `x, q` a numeric vector of quantiles. `p` a numeric vector of probabilities. `n` number of observations to return. `n1, n2` the number of elements of first and second type, respectively. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x].

## Details

The Runs distribution has probability function

P(R=r) = 2 choose(n1-1,r/2-1)choose(n2-1,r/2-1)/choose(n1+n2,n1), if r is even P(R=r) =

for r = 2, 3, …, 2 min(n1+n2)+c with c=0 if n1 = n2 or c=1 if n_1 =! n_2.

If an element of `x` is not integer, the result of `druns` is zero.

The quantile is defined as the smallest value x such that F(x) ≥ p, where F is the distribution function.

## Value

`druns` gives the probability function, `pruns` gives the distribution function and `qruns` gives the quantile function.

## References

Swed, F.S. and Eisenhart, C. (1943). Tables for Testing Randomness of Grouping in a Sequence of Alternatives, Ann. Math Statist. 14(1), 66-87.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```## ## Example: Distribution Function ## Creates Table I in Swed and Eisenhart (1943), p. 70, ## with n1 = 2 and n1 <= n2 <= 20 ## m <- NULL for (i in 2:20){ m <- rbind(m, pruns(2:5,2,i)) } rownames(m)=2:20 colnames(m)=2:5 # # 2 3 4 5 # 2 0.333333333 0.6666667 1.0000000 1 # 3 0.200000000 0.5000000 0.9000000 1 # 4 0.133333333 0.4000000 0.8000000 1 # 5 0.095238095 0.3333333 0.7142857 1 # 6 0.071428571 0.2857143 0.6428571 1 # 7 0.055555556 0.2500000 0.5833333 1 # 8 0.044444444 0.2222222 0.5333333 1 # 9 0.036363636 0.2000000 0.4909091 1 # 10 0.030303030 0.1818182 0.4545455 1 # 11 0.025641026 0.1666667 0.4230769 1 # 12 0.021978022 0.1538462 0.3956044 1 # 13 0.019047619 0.1428571 0.3714286 1 # 14 0.016666667 0.1333333 0.3500000 1 # 15 0.014705882 0.1250000 0.3308824 1 # 16 0.013071895 0.1176471 0.3137255 1 # 17 0.011695906 0.1111111 0.2982456 1 # 18 0.010526316 0.1052632 0.2842105 1 # 19 0.009523810 0.1000000 0.2714286 1 # 20 0.008658009 0.0952381 0.2597403 1 # ```

### Example output

```
```

randtests documentation built on May 2, 2019, 3:26 a.m.