Compute the test statistic of the Baumgartner-Weiss-Schindler test.
Compute the Baumgartner-Weiss-Schindler test statistic.
bws_stat(x, y)
x |
a vector. |
y |
a vector. |
Given vectors X and Y, computes B_X and B_Y as
described by Baumgartner et al., returning their average, B.
The test statistic approximates the variance-weighted square norm of the
difference in CDFs of the two distributions. For sufficiently large sample
sizes (more than 20, say), under the null the test statistic approaches the asymptotic
value computed in bws_cdf.
The test value is an approximation of
\tilde{B} = \frac{mn}{m+n} \int_0^1 \frac{1}{z(1-z)} ≤ft(F_X(z) - F_Y(z)\right)^2 \mathrm{dz},
where m (n) is the number of elements in X (Y), and F_X(z) (F_Y(z)) is the CDF of X (Y).
The test statistic is based only on the ranks of the input. If the same monotonic transform is applied to both vectors, the result should be unchanged. Moreover, the test is inherently two-sided, so swapping X and Y should also leave the test statistic unchanged.
The BWS test statistic, B.
Steven E. Pav shabbychef@gmail.com
W. Baumgartner, P. Weiss, H. Schindler, 'A nonparametric test for the general two-sample problem', Biometrics 54, no. 3 (Sep., 1998): pp. 1129-1135. http://doai.io/10.2307/2533862
set.seed(1234)
x <- runif(1000)
y <- runif(100)
bval <- bws_stat(x,y)
# check a monotonic transform:
ftrans <- function(x) { log(1 + x) }
bval2 <- bws_stat(ftrans(x),ftrans(y))
stopifnot(all.equal(bval,bval2))
# check commutivity
bval3 <- bws_stat(y,x)
stopifnot(all.equal(bval,bval3))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.