We discuss a parameter estimation problem for a Gaussian copula model under misspecification. Conventional estimators such as the maximum likelihood estimator (MLE) do not work well if the model is misspecified. We propose the estimator that minimizes the projective power entropy. We call it the
Applications of copula models have been increasing in number in recent years. There are a variety of applications on finance, risk management [
We make use of the
The
This paper is organized as follows. The
In Section
The density function of the Gaussian copula is given by
Let
We consider the MLE for the Gaussian copula model on the same setting as in Section
We give a fixed point algorithm to obtain the
(1) Set an appropriate correlation matrix
(2) Given
(3) For sufficient small given number
(4) For all local minimizers, repeat procedures 1–3 for different initial values
If we consider the estimation problem on Gaussian distributions with mean
We make a remark on the algorithm to obtain the MLE, or
In Section
The projective power entropy of
Let
In calculating the
We assume that
It is natural for us to require the equivalence of the two
The argument above extends to a general statement. For given one to one transformation
The
If
In this case we note that the
Next we consider the misspecification case where the true data generating process is given by (
In the case with dimension 2,
Suppose the true correlation matrices
Illustration of
Suppose
Like these examples,
So far we have considered the
In [
In this section we examine robustness of the
The
When
For example, if
This subsection describes the results of Monte Carlo simulations carried out in order to examine the robustness of the
RMSE of the norm for the

 

RMSE  0.155  0.808 
The property of the
We conducted two kinds of simulation.
The
We adopt the MLE for a mixture Gaussian copula model (
A set of data of size
Ratio of the number of success for the

500  1000 
 
Ratio  0.768  0.968 
Simulation 1. RMSE of the norm of




 

RMSE 
500  0.600  0.184  0.476  0.186 
1000  0.479  0.127  0.431  0.129 
Ratio of the number of success for the

500  1000 
 
Ratio  0.61  0.966 
Simulation 2. RMSE of the norm of




 

RMSE  500  0.494  0.946  0.563  0.952 
1000  0.468  1.010  0.438  1.032 
We have considered an estimation problem for misspecified Gaussian copula model. By the simulation study our methodology has been found to work well for misspecification. Though we did not consider how to determine the value of
We choose the measure in terms of invariance. However the
Another issue is to what extent the methodology here works for time series data. Because the basic premise of this paper is that we have data as quantiles, our method would fit, for example, the modeling of unconditional loss distribution [
We derive the estimation equation for
We see that
If
We see