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

 Runs R Documentation

## Distribution of the Wald Wolfowitz Runs Statistic

### Description

Probability function, distribution function, quantile function and random generation for the distribution of the Runs statistic obtained from samples with n_1 and n_2 elements of each type.

### Usage

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 \le x] otherwise, P[X > x].

### Details

The Runs distribution has probability function

 P(R=r)= \left\{ \begin{array}{cc} \frac{2{n_1-1 \choose r/2-1}{n_2-1 \choose r/2-1}}{{n_1+n_2 \choose n_1}}, & \mbox{if } r \mbox{ is even}\\ \frac{{n_1-1 \choose (r-1)/2}{n_2-1 \choose (r-3)/2}\,+\,{n_1-1 \choose (r-3)/2}{n_2-1 \choose (r-1)/2}}{{n_1+n_2 \choose n_1}}, & \mbox{if } r \mbox{ is odd}\\ \end{array} \right. %\qquad r=2,3,\ldots, n_1+n_2. 

for r=2,3,\ldots, 2\min(n_1+n_2)+c with c=0 if n_1=n_2 or c=1 if n_1 \neq 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) \ge 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

##
## 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
#


randtests documentation built on June 25, 2024, 1:15 a.m.