GOFmontecarlo {nsRFA}

Goodness of fit tests

Description

Anderson-Darling goodness of fit tests for Regional Frequency Analysis: Monte-Carlo method.

Usage

 gofNORMtest (x)
 gofEXPtest (x, Nsim=1000)
 gofGUMBELtest (x, Nsim=1000)
 gofGENLOGIStest (x, Nsim=1000)
 gofGENPARtest (x, Nsim=1000)
 gofGEVtest (x, Nsim=1000)
 gofLOGNORMtest (x, Nsim=1000)
 gofP3test (x, Nsim=1000)

Arguments

x

data sample

Nsim

number of simulated samples from the hypothetical parent distribution

Details

An introduction, analogous to the following one, on the Anderson-Darling test is available on https://en.wikipedia.org/wiki/Anderson-Darling_test.

Given a sample x_i \ (i=1,\ldots,m) of data extracted from a distribution F_R(x), the test is used to check the null hypothesis H_0 : F_R(x) = F(x,\theta), where F(x,\theta) is the hypothetical distribution and \theta is an array of parameters estimated from the sample x_i.

The Anderson-Darling goodness of fit test measures the departure between the hypothetical distribution F(x,\theta) and the cumulative frequency function F_m(x) defined as:

F_m(x) = 0 \ , \ x < x_{(1)}

F_m(x) = i/m \ , \ x_{(i)} \leq x < x_{(i+1)}

F_m(x) = 1 \ , \ x_{(m)} \leq x

where x_{(i)} is the i-th element of the ordered sample (in increasing order).

The test statistic is:

Q^2 = m \! \int_x \left[ F_m(x) - F(x,\theta) \right]^2 \Psi(x) \,dF(x)

where \Psi(x), in the case of the Anderson-Darling test (Laio, 2004), is \Psi(x) = [F(x,\theta) (1 - F(x,\theta))]^{-1}. In practice, the statistic is calculated as:

A^2 = -m -\frac{1}{m} \sum_{i=1}^m \left\{ (2i-1)\ln[F(x_{(i)},\theta)] + (2m+1-2i)\ln[1 - F(x_{(i)},\theta)] \right\}

The statistic A^2, obtained in this way, may be confronted with the population of the A^2's that one obtain if samples effectively belongs to the F(x,\theta) hypothetical distribution. In the case of the test of normality, this distribution is defined (see Laio, 2004). In other cases, e.g. the Pearson Type III case, can be derived with a Monte-Carlo procedure.

Value

gofNORMtest tests the goodness of fit of a normal (Gauss) distribution with the sample x.

gofEXPtest tests the goodness of fit of a exponential distribution with the sample x.

gofGUMBELtest tests the goodness of fit of a Gumbel (EV1) distribution with the sample x.

gofGENLOGIStest tests the goodness of fit of a Generalized Logistic distribution with the sample x.

gofGENPARtest tests the goodness of fit of a Generalized Pareto distribution with the sample x.

gofGEVtest tests the goodness of fit of a Generalized Extreme Value distribution with the sample x.

gofLOGNORMtest tests the goodness of fit of a 3 parameters Lognormal distribution with the sample x.

gofP3test tests the goodness of fit of a Pearson type III (gamma) distribution with the sample x.

They return the value A_2 of the Anderson-Darling statistics and its non exceedence probability P. Note that P is the probability of obtaining the test statistic A_2 lower than the one that was actually observed, assuming that the null hypothesis is true, i.e., P is one minus the p-value usually employed in statistical testing (see https://en.wikipedia.org/wiki/P-value). If P(A_2) is, for example, greater than 0.90, the null hypothesis at significance level \alpha=10\% is rejected.

Note

For information on the package and the Author, and for all the references, see nsRFA.

See Also

traceWminim, roi, HOMTESTS.

Examples

x <- rnorm(30,10,1)
gofNORMtest(x)

x <- rand.gamma(50, 100, 15, 7)
gofP3test(x, Nsim=200)

x <- rand.GEV(50, 0.907, 0.169, 0.0304)
gofGEVtest(x, Nsim=200)

x <- rand.genlogis(50, 0.907, 0.169, 0.0304)
gofGENLOGIStest(x, Nsim=200)

x <- rand.genpar(50, 0.716, 0.418, 0.476)
gofGENPARtest(x, Nsim=200)

x <- rand.lognorm(50, 0.716, 0.418, 0.476)
gofLOGNORMtest(x, Nsim=200)