cvm.test {dgof}

Discrete Cramer-von Mises Goodness-of-Fit Tests

Description

Computes the test statistics for doing one-sample Cramer-von Mises goodness-of-fit tests and calculates asymptotic p-values.

Usage

cvm.test(x, y, type = c("W2", "U2", "A2"),
         simulate.p.value=FALSE, B=2000, tol=1e-8)

Arguments

x

a numerical vector of data values.

y

an ecdf or step-function (stepfun) for specifying the hypothesized model.

type

the variant of the Cramer-von Mises test; "W2" is the default and most common method, "U2" is for cyclical data, and "A2" is the Anderson-Darling alternative. For details see references.

simulate.p.value

a logical indicating whether to compute p-values by Monte Carlo simulation.

B

an integer specifying the number of replicates used in the Monte Carlo test (for discrete goodness-of-fit tests only).

tol

used as an upper bound for possible rounding error in values (say, a and b) when needing to check for equality (a==b) (for discrete goodness-of-fit tests only).

Details

While the Kolmogorov-Smirnov test may be the most popular of the nonparametric goodness-of-fit tests, Cramer-von Mises tests have been shown to be more powerful against a large class of alternatives hypotheses. The original test was developed by Harald Cramer and Richard von Mises (Cramer, 1928; von Mises, 1928) and further adapted by Anderson and Darling (1952), and Watson (1961).

Value

An object of class htest.

Note

Additional notes?

Author(s)

Taylor B. Arnold and John W. Emerson

Maintainer: Taylor B. Arnold <[e-mail]>

References

T. W. Anderson and D. A. Darling (1952). Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. Annals of Mathematical Statistics, 23:193-212.

V. Choulakian, R. A. Lockhart, and M. A. Stephens (1994). Cramer-von Mises statistics for discrete distributions. The Canadian Journal of Statistics, 22(1): 125-137.

H. Cramer (1928). On the composition of elementary errors. Skand. Akt., 11:141-180.

M. A. Stephens (1974). Edf statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69(347): 730-737.

R. E. von Mises (1928). Wahrscheinlichkeit, Statistik und Wahrheit. Julius Springer, Vienna, Austria.

G. S. Watson (1961). Goodness of fit tests on the circle. Biometrika, 48:109-114.

See Also

ks.test, ecdf, stepfun

Examples

require(dgof)

x3 <- sample(1:10, 25, replace=TRUE)

# Using ecdf() to specify a discrete distribution:
ks.test(x3, ecdf(1:10))
cvm.test(x3, ecdf(1:10))

# Using step() to specify the same discrete distribution:
myfun <- stepfun(1:10, cumsum(c(0, rep(0.1, 10))))
ks.test(x3, myfun)
cvm.test(x3, myfun)

# Usage of U2 for cyclical distributions (note U2 unchanged, but W2 not)

set.seed(1)
y <- sample(1:4, 20, replace=TRUE)
cvm.test(y, ecdf(1:4), type='W2')
cvm.test(y, ecdf(1:4), type='U2')
z <- y
cvm.test(z, ecdf(1:4), type='W2')
cvm.test(z, ecdf(1:4), type = 'U2')

# Compare analytic results to simulation results
set.seed(1)
y <- sample(1:3, 10, replace=TRUE)
cvm.test(y, ecdf(1:6), simulate.p.value=FALSE)
cvm.test(y, ecdf(1:6), simulate.p.value=TRUE)