gofPIOSTn tests a 2 or 3 dimensional dataset with the PIOS test for a copula. The possible copulae are "normal", "t", "gumbel", "clayton" and "frank". The parameter estimation is performed with pseudo maximum likelihood method. In case the estimation fails, inversion of Kendall's tau is used. The approximate p-values are computed with a semiparametric bootstrap, which computation can be accelerated by enabling in-build parallel computation.
gofPIOSTn(copula, x, M = 1000, param = 0.5, param.est = T, df = 4, df.est = T, margins = "ranks", dispstr = "ex", m = 1, execute.times.comp = T, processes = 1)"normal", "t", "clayton", "gumbel" and "frank".
TRUE or FALSE. TRUE means that param will be estimated with a maximum likelihood estimation.
"t"-copula.
df shall be estimated. Has to be either FALSE or TRUE, where TRUE means that it will be estimated.
"ranks", which is the standard approach to convert data in such a case. Alternatively can the following distributions be specified: "beta", "cauchy", Chi-squared ("chisq"), "f", "gamma", Log normal ("lnorm"), Normal ("norm"), "t", "weibull", Exponential ("exp").
copula.
M is at least 100.
class gofCOP with the components
gofCOP with the componentsm out of the data. The test compares then the pseudo likelihood of the data in each block with the overall parameter and with the parameter by leaving out the data in the block. By this procedure can be determined if the data in the block influence the parameter estimation significantly. The test statistic is defined as
$$T = \sum_{b=1}^M \sum_{k=1}^m [l\{U_k^b;\theta_n \} - l\{U_k^b;\theta_n^{-b} \}]$$with the pseudo observations $U[ij]$ for $i = 1, ...,n$; $j = 1, ...,d$ and $$\theta_n = \arg \min_{\theta} \sum_{i=1}^n l(U_i; \theta)$$ and $$\theta_n^{-b} = \arg \min_{\theta} \sum_{b^{'} \neq b}^M \sum_{i=1}^m l(U_i^{b^{'}}; \theta), b=1, \dots, M.$$
The approximate p-value is computed by the formula $$\sum_{b=1}^M \mathbf{I}(|T_b| \geq |T|) / M,$$
The applied estimation method is the two-step pseudo maximum likelihood approach, see Genest and Rivest (1995).
For small values of M, initializing the parallization via processes does not make sense. The registration of the parallel processes increases the computation time. Please consider to enable parallelization just for high values of M.
data(IndexReturns)
gofPIOSTn("normal", IndexReturns[c(1:100),c(1:2)], M = 20)
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