GOFmontecarlo             package:nsRFA             R Documentation

_G_o_o_d_n_e_s_s _o_f _f_i_t _t_e_s_t_s

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

      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)

_A_r_g_u_m_e_n_t_s:

       x: data sample

    Nsim: number of simulated samples from the hypothetical parent
          distribution

_D_e_t_a_i_l_s:

     An introduction, analogous to the following one, on the
     Anderson-Darling test is available on <URL:
     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.

_V_a_l_u_e:

     '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 alpha=10%.

_N_o_t_e:

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

_S_e_e _A_l_s_o:

     'traceWminim', 'roi', 'HOMTESTS'.

_E_x_a_m_p_l_e_s:

     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)

