nsRFA (version 0.7-16)

GOFmontecarlo: 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)

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 $\alpha =10\mathrm{\%}$ is rejected.

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 $x_i(i=1,\dots ,m)$ 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: $$F_m(x)=0,x<x_{(1)}$$ $$F_m(x)=i/m,x_{(i)}\le x<x_{(i+1)}$$ $$F_m(x)=1,x_{(m)}\le x$$ where $x_{(i)}$ is the $i$-th element of the ordered sample (in increasing order).

The test statistic is: $$Q^2=m{\int }_x{[F_m(x)-F(x,\theta )]}^2\mathrm{\Psi }(x)dF(x)$$ where $\mathrm{\Psi }(x)$, in the case of the Anderson-Darling test (Laio, 2004), is $\mathrm{\Psi }(x)=[F(x,\theta )(1-F(x,\theta )){]}^{-1}$. In practice, the statistic is calculated as: $$A^2=-m-\frac{1}{m}\sum _{i=1}^m\{(2i-1)\ln [F(x_{(i)},\theta )]+(2m+1-2i)\ln [1-F(x_{(i)},\theta )]\}$$

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.

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)