Package 'goftest'

Classical Goodness-of-Fit Tests

Description

Cramer-von Mises and Anderson-Darling tests of goodness-of-fit for continuous univariate distributions, using modern algorithms to compute the null distributions.

Details

The goftest package contains implementations of the classical Cramer-von Mises and Anderson-Darling tests of goodness-of-fit for continuous univariate distributions.

The Cramer-von Mises test is performed by cvm.test. The cumulative distribution function of the null distribution of the test statistic is computed by pCvM using the algorithm of Csorgo and Faraway (1996). The quantiles are computed by qCvM by root-finding.

The Anderson-Darling test is performed by ad.test. The cumulative distribution function of the null distribution of the test statistic is computed by pAD using the algorithm of Marsaglia and Marsaglia (2004). The quantiles are computed by qAD by root-finding.

By default, each test assumes that the parameters of the null distribution are known (a simple null hypothesis). If the parameters were estimated (calculated from the data) then the user should set estimated=TRUE which uses the method of Braun (1980) to adjust for the effect of estimating the parameters from the data.

Author(s)

Adrian Baddeley, Julian Faraway, John Marsaglia, George Marsaglia.

Maintainer: Adrian Baddeley <[email protected]>

References

Braun, H. (1980) A simple method for testing goodness-of-fit in the presence of nuisance parameters. Journal of the Royal Statistical Society 42, 53–63.

Csorgo, S. and Faraway, J.J. (1996) The exact and asymptotic distributions of Cramer-von Mises statistics. Journal of the Royal Statistical Society, Series B 58, 221–234.

Marsaglia, G. and Marsaglia, J. (2004) Evaluating the Anderson-Darling Distribution. Journal of Statistical Software 9 (2), 1–5. February 2004. http://www.jstatsoft.org/v09/i02

See Also

ks.test

Examples

x <- rnorm(30, mean=2, sd=1)
  # default behaviour: parameters fixed: simple null hypothesis
  cvm.test(x, "pnorm", mean=2, sd=1)
  ad.test(x, "pnorm", mean=2, sd=1)
  # parameters estimated: composite null hypothesis
  mu <- mean(x)
  sigma <- sd(x)
  cvm.test(x, "pnorm", mean=mu, sd=sigma, estimated=TRUE)
  ad.test(x, "pnorm", mean=mu, sd=sigma, estimated=TRUE)

---

Anderson-Darling Test of Goodness-of-Fit

Description

Performs the Anderson-Darling test of goodness-of-fit to a specified continuous univariate probability distribution.

Usage

ad.test(x, null = "punif", ..., estimated=FALSE, nullname)

Arguments

x	Numeric vector of data values.
null	A function, or a character string giving the name of a function, to compute the cumulative distribution function for the null distribution.
...	Additional arguments for the cumulative distribution function.
estimated	Logical value indicating whether the parameters of the distribution were estimated using the data x (composite null hypothesis), or were fixed in advance (simple null hypothesis, the default).
nullname	Optional character string describing the null distribution. The default is "uniform distribution".

Details

This command performs the Anderson-Darling test of goodness-of-fit to the distribution specified by the argument null. It is assumed that the values in x are independent and identically distributed random values, with some cumulative distribution function $F$. The null hypothesis is that $F$ is the function specified by the argument null, while the alternative hypothesis is that $F$ is some other function.

By default, the test assumes that all the parameters of the null distribution are known in advance (a simple null hypothesis). This test does not account for the effect of estimating the parameters.

If the parameters of the distribution were estimated (that is, if they were calculated from the same data x), then this should be indicated by setting the argument estimated=TRUE. The test will then use the method of Braun (1980) to adjust for the effect of parameter estimation.

Note that Braun's method involves randomly dividing the data into two equally-sized subsets, so the $p$-value is not exactly the same if the test is repeated. This technique is expected to work well when the number of observations in x is large.

Value

An object of class "htest" representing the result of the hypothesis test.

Author(s)

Original C code by George Marsaglia and John Marsaglia. R interface by Adrian Baddeley.

References

Anderson, T.W. and Darling, D.A. (1952) Asymptotic theory of certain 'goodness-of-fit' criteria based on stochastic processes. Annals of Mathematical Statistics 23, 193–212.

Anderson, T.W. and Darling, D.A. (1954) A test of goodness of fit. Journal of the American Statistical Association 49, 765–769.

Braun, H. (1980) A simple method for testing goodness-of-fit in the presence of nuisance parameters. Journal of the Royal Statistical Society 42, 53–63.

Marsaglia, G. and Marsaglia, J. (2004) Evaluating the Anderson-Darling Distribution. Journal of Statistical Software 9 (2), 1–5. February 2004. http://www.jstatsoft.org/v09/i02

See Also

pAD for the null distribution of the test statistic.

Examples

x <- rnorm(10, mean=2, sd=1)
ad.test(x, "pnorm", mean=2, sd=1)
ad.test(x, "pnorm", mean=mean(x), sd=sd(x), estimated=TRUE)

---

Cramer-Von Mises Test of Goodness-of-Fit

Description

Performs the Cramer-von Mises test of goodness-of-fit to a specified continuous univariate probability distribution.

Usage

cvm.test(x, null = "punif", ..., estimated=FALSE, nullname)

Arguments

x	Numeric vector of data values.
null	A function, or a character string giving the name of a function, to compute the cumulative distribution function for the null distribution.
...	Additional arguments for the cumulative distribution function.
estimated	Logical value indicating whether the parameters of the distribution were estimated using the data x (composite null hypothesis), or were fixed in advance (simple null hypothesis, the default).
nullname	Optional character string describing the null distribution. The default is "uniform distribution".

Details

This command performs the Cramer-von Mises test of goodness-of-fit to the distribution specified by the argument null. It is assumed that the values in x are independent and identically distributed random values, with some cumulative distribution function $F$. The null hypothesis is that $F$ is the function specified by the argument null, while the alternative hypothesis is that $F$ is some other function.

By default, the test assumes that all the parameters of the null distribution are known in advance (a simple null hypothesis). This test does not account for the effect of estimating the parameters.

If the parameters of the distribution were estimated (that is, if they were calculated from the same data x), then this should be indicated by setting the argument estimated=TRUE. The test will then use the method of Braun (1980) to adjust for the effect of parameter estimation.

Note that Braun's method involves randomly dividing the data into two equally-sized subsets, so the $p$-value is not exactly the same if the test is repeated. This technique is expected to work well when the number of observations in x is large.

Value

An object of class "htest" representing the result of the hypothesis test.

Author(s)

Adrian Baddeley.

References

Braun, H. (1980) A simple method for testing goodness-of-fit in the presence of nuisance parameters. Journal of the Royal Statistical Society 42, 53–63.

Csorgo, S. and Faraway, J.J. (1996) The exact and asymptotic distributions of Cramer-von Mises statistics. Journal of the Royal Statistical Society, Series B 58, 221–234.

See Also

pCvM for the null distribution of the test statistic.

Examples

x <- rnorm(10, mean=2, sd=1)
cvm.test(x, "pnorm", mean=2, sd=1)
cvm.test(x, "pnorm", mean=mean(x), sd=sd(x), estimated=TRUE)

---

Null Distribution of Anderson-Darling Test Statistic

Description

pAD computes the cumulative distribution function, and qAD computes the quantile function, of the null distribution of the Anderson-Darling test statistic.

Usage

pAD(q, n = Inf, lower.tail = TRUE, fast=TRUE)
  qAD(p, n = Inf, lower.tail = TRUE, fast=TRUE)

Arguments

q	Numeric vector of quantiles (values for which the cumulative probability is required).
p	Numeric vector of probabilities.
n	Integer. Sample size for the Anderson-Darling test.
lower.tail	Logical. If TRUE (the default), probabilities are $P(X \le q)$, and otherwise they are $P(X > q)$.
fast	Logical value indicating whether to use a fast algorithm or a slower, more accurate algorithm, in the case n=Inf.

Details

pAD uses the algorithms and C code described in Marsaglia and Marsaglia (2004).

qAD uses uniroot to find the quantiles.

The argument fast applies only when n=Inf and determines whether the asymptotic distribution is approximated using the faster algorithm adinf (accurate to 4-5 places) or the slower algorithm ADinf (accurate to 11 places) described in Marsaglia and Marsaglia (2004).

Value

A numeric vector of the same length as p or q.

Author(s)

Original C code by G. and J. Marsaglia. R interface by Adrian Baddeley.

References

Anderson, T.W. and Darling, D.A. (1952) Asymptotic theory of certain 'goodness-of-fit' criteria based on stochastic processes. Annals of Mathematical Statistics 23, 193–212.

Anderson, T.W. and Darling, D.A. (1954) A test of goodness of fit. Journal of the American Statistical Association 49, 765–769.

Marsaglia, G. and Marsaglia, J. (2004) Evaluating the Anderson-Darling Distribution. Journal of Statistical Software 9 (2), 1–5. February 2004. http://www.jstatsoft.org/v09/i02

See Also

ad.test

Examples

pAD(1.1, n=5)
  pAD(1.1)
  pAD(1.1, fast=FALSE)

  qAD(0.5, n=5)
  qAD(0.5)

---

Null Distribution of Cramer-von Mises Test Statistic

Description

pCvM computes the cumulative distribution function, and qCvM computes the quantile function, of the null distribution of the Cramer-von Mises test statistic.

Usage

pCvM(q, n = Inf, lower.tail = TRUE)
  qCvM(p, n = Inf, lower.tail = TRUE)

Arguments

q	Numeric vector of quantiles (values for which the cumulative probability is required).
p	Numeric vector of probabilities.
n	Integer. Sample size for the Cramer-von Mises test.
lower.tail	Logical. If TRUE (the default), probabilities are $P(X \le q)$, and otherwise they are $P(X > q)$.

Details

For finite n the cumulative distribution function is approximated by the first order expansion $V(x) + \psi_1(x)/n$, equation (1.8) of Csorgo and Faraway (1996).

qCvM uses uniroot to find the quantiles.

Value

A numeric vector of the same length as p or q.

Author(s)

Original Matlab code by Julian Faraway, translated to R by Adrian Baddeley.

References

Csorgo, S. and Faraway, J.J. (1996) The exact and asymptotic distributions of Cramer-von Mises statistics. Journal of the Royal Statistical Society, Series B 58, 221–234.

See Also

cvm.test

Examples

pCvM(1.1, n=5)
  pCvM(1.1)

  qCvM(0.5, n=5)
  qCvM(0.5)

---

Explanatory Name of Distribution Function

Description

Recognises many standard cumulative distribution functions and returns a string describing the distribution.

Usage

recogniseCdf(s="punif")

Arguments

s	A single character string giving the name of an R function that calculates cumulative probabilities.

Details

The list of recognised distribution functions includes all those available in the stats package and in goftest.

By convention, the name of a cumulative distribution function begins with the letter p. For example, punif is the cumulative distribution function of the uniform distribution.

The initial letter p can be omitted in the function recogniseCdf.

Value

Character string, or NULL if the name is not recognised.

Author(s)

Adrian Baddeley.

See Also

pAD

Examples

recogniseCdf("punif")
   recogniseCdf("unif")
   recogniseCdf("pt")

---