gofCopula (version 0.1-2)

gofPIOSRn: 2 dimensional gof test based on the in-and-out-of-sample approach

Description

gofPIOSRn tests a 2 dimensional dataset with the approximate PIOS test for a copula. The possible copulae are "gaussian", "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.

Usage

gofPIOSRn(copula, x, M = 1000, param = 0.5, param.est = T, df = 4, df.est = T, 
          margins = "ranks", execute.times.comp = T)

Arguments

copula

The copula to test for. Possible are the copulae "gaussian", "t", "clayton", "gumbel" and "frank".

x

A 2 dimensional matrix containing the residuals of the data.

M

Number of bootstrapping loops.

param

The parameter to be used.

param.est

Shall be either TRUE or FALSE. TRUE means that param will be estimated with a maximum likelihood estimation.

df

Degrees of freedom, if not meant to be estimated. Only necessary if tested for "t"-copula.

df.est

Indicates if df shall be estimated. Has to be either FALSE or TRUE, where TRUE means that it will be estimated.

margins

Specifies which estimation method shall be used in case that the input data are not in the range [0,1]. The default is "ranks", which is the standard approach to convert data in such a case. Alternatively can the following distributions be sp

execute.times.comp

Logical. Defines if the time which the estimation most likely takes shall be computed. It'll be just given if M is at least 100.

Value

• A object of the class gofCOP with the components
• method
  a character which informs about the performed analysis
• statistic
  value of the test statistic
• p.value
  the approximate p-value

Details

The "Rn" test is introduced in Zhang et al. (2015). It is a information ratio statistic which is approximately equivalent to the "Tn" test, which is the PIOS test. Both test the $H_0$ hypothesis $$H_0 : C_0 \in \mathcal{C}.$$ "Rn" is introduced because the "Tn" test has to estimate $n/m$ parameters which can be computationally demanding. The test statistic of the "Tn" test is defined as $$T = \sum_{b=1}^B \sum_{i=1}^m [l{U_i^b;\theta_n } - l{U_i^b;\theta_n^{-b} }]$$ with the pseudo observations $U_{ij}$ for $i = 1, \dots,n$; $j = 1, \dots,d$ and $$\theta_n = \arg \min_{\theta} \sum_{i=1}^n l(U_i; \theta)$$ and $$\theta_n^{-b} = \arg \min_{\theta} \sum_{b^{'} \neq b}^B \sum_{i=1}^m l(U_i^{b^{'}}; \theta), b=1, \dots, B.$$ By defining two information matrices $$S(\theta) = - E_0 [\frac{\partial^2}{\partial \theta \partial \theta^{\top}}l {U_1; \theta } ],$$ $$V(\theta) = - E_0 [\frac{\partial}{\partial \theta} l {U_1; \theta } l^{\top} {U_1; \theta } ]$$ where $S(\cdot)$ represents the negative sensitivity matrix, $V(\cdot)$ the variability matrix and $E_0$ is the expectation under the true copula $C_0$. Under suitable regularity conditions, given in Zhang et al. (2015), holds then in probability, that $$T = tr{S(\theta^{*})^{-1} - V(\theta^{*} }$$ as $n \rightarrow \infty.$ The approximate p-value is computed by the formula $$p = \frac{1}{B} \sum_{b = 1}^B \mathbf{I}(|T_b| \geq T).$$ For more details, see Zhang et al. (2015). The applied estimation method is the two-step pseudo maximum likelihood approach, see Genest and Rivest (1995).

References

Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit Test For Specification of Semiparametric Copula Dependence Models. under revision in Journal of Econometrics from 15.01.2014 http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2013-041.pdf Genest, C., K. G. and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82:534-552

Examples

data = cbind(rnorm(100), rnorm(100))

gofPIOSRn("gaussian", data, M = 20)