R: Distribution of the Smirnov Statistic

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 `"two.sided"` (default), `"less"`, or `"greater"` indicating whether absolute (two-sided, default) or raw (one-sided) differences of frequencies define the test statistic. See ‘Details’.
`exact`	`NULL` or a logical indicating whether the exact (conditional on the pooled data values in `z`) distribution or the asymptotic distribution should be used.
`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 `TRUE` (default), probabilities are `P[D < q]`, otherwise, `P[D \ge q]`.
`log.p`	a logical, if `TRUE` (default), probabilities are given as log-probabilities.
`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 two-sided case,

D^+ = \sup_c ( F_{x,n_x}(c) - F_{y,n_y}(c) )

in the one-sided "greater" case, and

D^- = \sup_c ( F_{y,n_y}(c) - F_{x,n_x}(c) )

in the one-sided "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