Runs {randtests} | 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 |
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
#