Anderson-Darling goodness of fit tests for Regional Frequency Analysis: Monte-Carlo method.
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)data sample
number of simulated samples from the hypothetical parent distribution
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 http://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.
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 \(x_i \ (i=1,\ldots,m)\) 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.
# NOT RUN {
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)
# }
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