# Goodness of fit tests

### Description

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

### Usage

1 2 3 4 5 6 7 8 | ```
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,θ)*, where *F(x,θ)* is the hypothetical distribution and *θ* 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,θ)* 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,θ)]^2 Ψ(x) dF(x)*

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

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

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,θ)* 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 non exceedence probability *P*.
Note that *P* is the probability of obtaining the test statistic *A2* 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 http://en.wikipedia.org/wiki/P-value).
If *P(A2)* is, for example, greater than 0.90, the null hypothesis at significance level *α=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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 | ```
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)
``` |