GOFmontecarlo {nsRFA} | R Documentation

## 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 http://en.wikipedia.org/wiki/Anderson-Darling_test.

Given a sample xi (i=1,...,m) of data extracted from a distribution FR(x), the test is used to check the null hypothesis H0 : FR(x) = F(x,theta), where F(x,theta) is the hypothetical distribution and theta is an array of parameters estimated from the sample xi.

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

Fm(x)=0, x<x(1)

Fm(x)=i/m, x(i)<=x<x(i+1)

Fm(x)=1, x(m)<=x

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

The test statistic is:

Q2 = m int[Fm(x) - F(x,theta)]^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:

A2 = -m -1/m sum{(2i-1)ln[F(x(i),theta)] + (2m+1-2i)ln[1 - F(x(i),theta)]}

The statistic A2, obtained in this way, may be confronted with the population of the A2'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 A2 of the Anderson-Darling statistics and its probability P. If P(A2) is, for example, greater than 0.90, the test is not passed at level α=10%.

### 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)

[Package nsRFA version 0.6-9 Index]