| gofPIOSRn {gofCopula} | R Documentation |
2 and 3 dimensional gof test based on the in-and-out-of-sample approach
Description
gofPIOSRn tests a 2 or 3 dimensional dataset with the approximate
PIOS test for a copula. The possible copulae are "normal",
"t", "clayton", "gumbel", "frank", "joe",
"amh", "galambos", "fgm" and "plackett". 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.
Usage
gofPIOSRn(
copula = c("normal", "t", "clayton", "gumbel", "frank", "joe", "amh", "galambos",
"fgm", "plackett"),
x,
param = 0.5,
param.est = TRUE,
df = 4,
df.est = TRUE,
margins = "ranks",
flip = 0,
M = 1000,
dispstr = "ex",
lower = NULL,
upper = NULL,
seed.active = NULL,
processes = 1
)
Arguments
copula |
The copula to test for. Possible are |
x |
A matrix containing the data with rows being observations and columns being variables. |
param |
The parameter to be used. |
param.est |
Shall be either |
df |
Degrees of freedom, if not meant to be estimated. Only necessary
if tested for |
df.est |
Indicates if |
margins |
Specifies which estimation method for the margins shall be
used. The default is |
flip |
The control parameter to flip the copula by 90, 180, 270 degrees clockwise. Only applicable for bivariate copula. Default is 0 and possible inputs are 0, 90, 180, 270 and NULL. |
M |
Number of bootstrapping loops. |
dispstr |
A character string specifying the type of the symmetric
positive definite matrix characterizing the elliptical copula. Implemented
structures are "ex" for exchangeable and "un" for unstructured, see package
|
lower |
Lower bound for the maximum likelihood estimation of the copula
parameter. The constraint is also active in the bootstrapping procedure. The
constraint is not active when a switch to inversion of Kendall's tau is
necessary. Default |
upper |
Upper bound for the maximum likelihood estimation of the copula
parameter. The constraint is also active in the bootstrapping procedure. The
constraint is not active when a switch to inversion of Kendall's tau is
necessary. Default |
seed.active |
Has to be either an integer or a vector of M+1 integers.
If an integer, then the seeds for the bootstrapping procedure will be
simulated. If M+1 seeds are provided, then these seeds are used in the
bootstrapping procedure. Defaults to |
processes |
The number of parallel processes which are performed to speed up the bootstrapping. Shouldn't be higher than the number of logical processors. Please see the details. |
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}^M
\sum_{k=1}^m \{l(U_k^b;\theta_n ) - l(U_k^b;\theta_n^{-b} )\}
with l the log likelihood function, 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}^M \sum_{k=1}^m
l(U_k^{b^{'}}; \theta), b=1, \dots, M.
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)
\{\frac{\partial}{\partial \theta} l (U_1; \theta) \}^{\top} ]
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
\sum_{b=1}^M
\mathbf{I}(|T_b| \geq |T|) / M,
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).
For small values of M, initializing the parallelisation via
processes does not make sense. The registration of the parallel
processes increases the computation time. Please consider to enable
parallelisation just for high values of M.
Value
An object of the class gofCOP with the components
method |
a character which informs about the performed analysis |
copula |
the copula tested for |
margins |
the method used to estimate the margin distribution. |
param.margins |
the parameters of
the estimated margin distributions. Only applicable if the margins were not
specified as |
theta |
dependence parameters of the copulae |
df |
the degrees of freedem of the copula. Only applicable for t-copula. |
res.tests |
a matrix with the p-values and test statistics of the hybrid and the individual tests |
References
Zhang, S., Okhrin, O., Zhou, Q., and Song, P.. Goodness-of-fit
Test For Specification of Semiparametric Copula Dependence Models.
Journal of Econometrics, 193, 2016, pp. 215-233
doi: 10.1016/j.jeconom.2016.02.017
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(IndexReturns2D)
gofPIOSRn("normal", IndexReturns2D, M = 10)