Smirnov {stats}  R Documentation 
Distribution of the Smirnov Statistic
Description
Distribution function, quantile function and random generation for the distribution of the Smirnov statistic.
Usage
psmirnov(q, sizes, z = NULL,
alternative = c("two.sided", "less", "greater"),
exact = TRUE, simulate = FALSE, B = 2000,
lower.tail = TRUE, log.p = FALSE)
qsmirnov(p, sizes, z = NULL,
alternative = c("two.sided", "less", "greater"),
exact = TRUE, simulate = FALSE, B = 2000)
rsmirnov(n, sizes, z = NULL,
alternative = c("two.sided", "less", "greater"))
Arguments
q 
a numeric vector of quantiles. 
p 
a numeric vector of probabilities. 
sizes 
an integer vector of length two giving the sample sizes. 
z 
a numeric vector of the pooled data values in both samples when the exact conditional distribution of the Smirnov statistic given the data shall be computed. 
alternative 
one of 
exact 

simulate 
a logical indicating whether to compute the distribution function by Monte Carlo simulation. 
B 
an integer specifying the number of replicates used in the Monte Carlo test. 
lower.tail 
a logical, if 
log.p 
a logical, if 
n 
an integer giving number of observations. 
Details
For samples x
and y
with respective sizes n_x
and
n_y
and empirical cumulative distribution functions
F_{x,n_x}
and F_{y,n_y}
, the Smirnov statistic is
D = \sup_c  F_{x,n_x}(c)  F_{y,n_y}(c) 
in the twosided case,
D^+ = \sup_c ( F_{x,n_x}(c)  F_{y,n_y}(c) )
in the onesided "greater"
case, and
D^ = \sup_c ( F_{y,n_y}(c)  F_{x,n_x}(c) )
in the onesided "less"
case.
These statistics are used in the Smirnov test of the null that x
and y
were drawn from the same distribution, see
ks.test
.
If the underlying common distribution function F
is continuous,
the distribution of the test statistics does not depend on F
,
and has a simple asymptotic approximation. For arbitrary F
, one
can compute the conditional distribution given the pooled data values
z
of x
and y
, either exactly (feasible provided that
the product n_x n_y
of the sample sizes is “small enough”) or
approximately Monte Carlo simulation. If the pooled data values z
are not specified, a pooled sample without ties is assumed.
Value
psmirnov
gives the distribution function,
qsmirnov
gives the quantile function, and
rsmirnov
generates random deviates.
See Also
ks.test
for references on the algorithms used for
computing exact distributions.