\Header{ks.test}{Kolmogorov-Smirnov Tests}
\keyword{htest}{ks.test}
\begin{Description}\relax
Performs one or two sample Kolmogorov-Smirnov tests.
\end{Description}
\begin{Usage}
\begin{verbatim}
ks.test(x, y, ..., alternative = c("two.sided", "less", "greater"),
        exact = NULL)
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] a numeric vector of data values.
\item[\code{y}] either a numeric vector of data values, or a character string
naming a distribution function.
\item[\code{...}] parameters of the distribution specified (as a character
string) by \code{y}.
\item[\code{alternative}] indicates the alternative hypothesis and must be
one of \code{"two.sided"} (default), \code{"less"}, or
\code{"greater"}.  You can specify just the initial letter.
\item[\code{exact}] \code{NULL} or a logical indicating whether an exact
p-value should be computed.  See Details for the meaning of \code{NULL}.
Only used in the two-sided two-sample case.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
If \code{y} is numeric, a two-sample test of the null hypothesis
that \code{x} and \code{y} were drawn from the same \emph{continuous}
distribution is performed.

Alternatively, \code{y} can be a character string naming a continuous
distribution function.  In this case, a one-sample test is carried
out of the null that the distribution function which generated
\code{x} is distribution \code{y} with parameters specified by \code{...}.

The presence of ties generates a warning, since continuous distributions
do not generate them.

The possible values \code{"two.sided"}, \code{"less"} and
\code{"greater"} of \code{alternative} specify the null hypothesis
that the true distribution function of \code{x} is equal to, not less
than or not greater than the hypothesized distribution function
(one-sample case) or the distribution function of \code{y} (two-sample
case), respectively.

Exact p-values are only available for the two-sided two-sample test
with no ties.  In that case, if \code{exact = NULL} (the default) an
exact p-value is computed if the product of the sample sizes is less
than 10000.  Otherwise, asymptotic distributions are used whose
approximations may be inaccurate in small samples.

If a single-sample test is used, the parameters specified in
\code{...} must be pre-specified and not estimated from the data.
There is some more refined distribution theory for the KS test with
estimated parameters (see Durbin, 1973), but that is not implemented
in \code{ks.test}.
\end{Details}
\begin{Value}
A list with class \code{"htest"} containing the following components:
\begin{ldescription}
\item[\code{statistic}] the value of the test statistic.
\item[\code{p.value}] the p-value of the test.
\item[\code{alternative}] a character string describing the alternative
hypothesis.
\item[\code{method}] a character string indicating what type of test was
performed.
\item[\code{data.name}] a character string giving the name(s) of the data.
\end{ldescription}
\end{Value}
\begin{References}\relax
William J. Conover (1971),
\emph{Practical nonparametric statistics}.
New York: John Wiley \& Sons.
Pages 295--301 (one-sample \dQuote{Kolmogorov} test),
309--314 (two-sample \dQuote{Smirnov} test).

Durbin, J. (1973)
\emph{Distribution theory for tests based on the sample distribution
function}.  SIAM.
\end{References}
\begin{SeeAlso}\relax
\code{\Link{shapiro.test}} which performs the Shapiro-Wilk test for
normality.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
x <- rnorm(50)
y <- runif(30)
# Do x and y come from the same distribution?
ks.test(x, y)
# Does x come from a shifted gamma distribution with shape 3 and scale 2?
ks.test(x+2, "pgamma", 3, 2) # two-sided
ks.test(x+2, "pgamma", 3, 2, alternative = "gr")
\end{ExampleCode}
\end{Examples}

