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\title{Simulation and Goodness-of-fit Test of Copulas}
\author{Yiran Chen, Major Prof: Giray \"{O}kten}
\institute{Applied and Computational  Mathematics, Department of Mathematics, Florida State University}
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\begin{document}
\begin{frame}[t]
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      \begin{block}{Introduction}
        \quad A \textbf{copula} is a multivariate distribution with dependent variables. It separates the dependence structure from the marginals. Copulas have been used widely in quantitative finance and insurance, such as option pricing and risk management.

        \quad Simulation plays a fundamental role in applications. There are two types of simulation methods for copulas, which can be compared through goodness-of-fit tests.

      \end{block}

      \vskip2ex

      \begin{block}{Simulation Methods}
        Sampling procedures for a $d$-dimensional copula $C$ can be viewed as a transformation
        \[
        \phi_C:[0,1]^k\rightarrow[0,1]^d \qquad
        \textup{i.e.}\qquad  \phi_C(\textbf{\textit{U}})= \textbf{\textit{V}}
        \]
       for some $k\geq d$.
        \begin{enumerate}
         \item \textbf{Stochastic Representation} writes a random vector in terms of other random vectors we know how to simulate efficiently. For example, if a random vector $\textbf{\textit{X}}\sim t_d(\mu,P,\nu)$, then it has the following representation
                \[
                \textbf{\textit{X}} \,{\buildrel d \over =}\, \mu+P^{1/2}\frac{\sqrt{\nu}}{\sqrt{\chi_\nu^2}}\textbf{\textit{Z}}.
                \]

         \vspace{.3cm}
         \item \textbf{Conditional Distribution Method} is given by conditional distribution transformation: for $\textbf{\textit{U}}\sim$ U$[0,1]^d$,
                \begin{eqnarray*}
                \left\{\begin{array}{ccc}
                V_1&=&U_1\\
                V_2&=&C^{-1}(U_2|V_1)\\
                &\dots&\\
                V_d&=&C^{-1}(U_d|V_1,\dots,V_{d-1})
                \end{array}\right.
                \end{eqnarray*}
             Then $\textbf{\textit{V}}=(V_1,\dots,V_d)$ is a sample from copula $C$.
         \end{enumerate}
         \begin{algorithm}{\textbf{A General Simulation Algorithm Using the CDM}}\label{Step}

             \quad\textbf{i.} Simulate $u_1$, $q_2$, $q_3$, $\dots$, $q_d$ independently from $\textbf{U}(0,1)$.

             \quad\textbf{ii.} Set $u_k=C_k^{-1}(q|u_1,\dots,u_{k-1})$,
            for $k=2,3,\dots, d$.

            Then vector $(u_1,\dots,u_d)$ is the desired sample from $C$.
         \end{algorithm}
      \end{block}

      \begin{block}{Monte Carlo vs Quasi-Monte Carlo}
      1000 realizations of two-dimensional pseudo-random number sequence (left) and low-discrepancy sequence (right):
      \begin{figure}[H]
        \begin{minipage}[t]{0.49\linewidth}
        \centering
        \includegraphics[width=4in]{MC.jpg}
        \end{minipage}
        \begin{minipage}[t]{0.49\linewidth}
        \centering
        \includegraphics[width=4in]{QMC.jpg}
        \end{minipage}
        \end{figure}
      \end{block}

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    \begin{block}{Examples for Bivariate Copulas}
    \textbf{Case 1:} 1000 realizations of a $t$ copula with three degree of freedom and correlation parameter $\rho=1/\sqrt{2}$, generated by a pseudo-random generator (left) and by a quasi-random number generator (right).
        \begin{figure}[H]
        \begin{minipage}[t]{0.49\linewidth}
        \centering
        \includegraphics[width=4.9in]{Figure5_top.jpg}
        \end{minipage}
        \begin{minipage}[t]{0.49\linewidth}
        \centering
        \includegraphics[width=4.9in]{Figure5_bottom.jpg}
        \end{minipage}
        \end{figure}
    \textbf{Case 2:} 1000 realizations of a Clayton copula with $\theta=2$, generated by a pseudo-random number generator (left) and by a quasi-random number generator (right).

        \begin{figure}[H]
        \begin{minipage}[t]{0.48\linewidth}
        \centering
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        \end{minipage}
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        \centering
        \includegraphics[width=4.9in]{Figure6_bottom.jpg}
        \end{minipage}
        \end{figure}
    \end{block}


\begin{block}{Goodness-of-fit Test}
Let $C$ be the underlying $d$-variate copula. Suppose one wants to test the goodness-of-fit (GoF) hypothesis that $C$ belongs to a copula family $C_\theta$ with parameter $\theta\in\Theta$, where $\Theta$ is the parameter space:
\[
\mathcal{H}_0: C\in\mathcal{C}=\{C_\theta;\theta\in\Theta\}
\qquad\textbf{vs.}\qquad
\mathcal{H}_1: C\notin\mathcal{C}=\{C_\theta;\theta\in\Theta\}.
\]

\begin{enumerate}
 \item \textbf{Rank data:} Suppose we have $N$ samples of the $d$-variate vector $\textbf{\textit{X}}_j=(X_{j,1},\dots,X_{j,d})$, for $j=1,\dots,N$. The rank data $\textbf{\textit{Z}}_j$ are given by
    \[
    \textbf{\textit{Z}}_j=(Z_{j,1},\dots,Z_{j,d})
    =\Big(\frac{R_{j,1}}{N+1},\dots, \frac{R_{j,d}}{N+1}\Big),\quad j=1,\dots,N;
    \]
    where $R_{j,i}$ is the rank of $X_{j,i}$ amongst $(X_{1,i},\dots,X_{N,i})$.

 \item \textbf{Conditional Probability Integral Transform (CPIT):} Let $\textbf{\textit{Z}}=(Z_1,\dots,Z_d)$ denote a random vector with marginal distributions $F_i(z_i)$ and conditional distribution $F_{i|1\dots i-1}(Z_i\leq z_i|Z_1=z_1,\dots,Z_{i-1}=z_{i-1})$ for $i=1,\dots,d$. \textbf{CPIT} of \textbf{\textit{Z}} is given by
    \begin{eqnarray*}
    V_1&=&\mathbb{P}(Z_1\leq z_1)=F_1(z_1),\\
    V_2&=&\mathbb{P}(Z_2\leq z_2|Z_1=z_1) =F_{2|1}(z_2|z_1),\\
    &\vdots&\\
    V_d&=&\mathbb{P}(Z_d\leq z_d |Z_1=z_1,\dots, Z_{d-1}=z_{d-1})=F_{d|1\dots d-1} (z_d|z_1,\dots,z_{d-1}).
    \end{eqnarray*}
    Then $\textbf{\textit{V}}=(V_1,\dots,V_d)$ has components that are i.i.d. \textbf{U}$(0,1)$ distributed.

 \end{enumerate}

\end{block}
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\begin{block}{CPIT Approach}
One GoF test is designed based on the CPIT:
\newline

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\put(0,78){\framebox(160,8)[l]{\quad $\{\textbf{X}_j\}_{j=1}^N$: \quad $N$ samples of {$\textbf{X}_j=(X_{j,1}, \dots, X_{j,d})$ from $C_{\theta}$}}}
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\put(12,58){Apply CPIT on each $\textbf{Z}_j$}

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\put(10,47){\vector(0,-1){19}}
\put(12,41){Dimension reduction by transformation function $T(x)=\Phi^{-1}(x)^2$:}
\put(15,32){$W_{B_j}=\sum\limits_{i=1}^dT(V_{j,i}) =\sum\limits_{i=1}^d\Phi^{-1}(V_{j,i})^2$}

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\put(12,13){Apply Anderson-Darling test on $\{W_{B_j}\}_{j=1}^N$}

\put(0,2){\framebox(160,8)[l]{\quad  $\widehat{\mathcal{T}}$: \quad AD Statistic with known distribution}}
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\begin{block}{Numerical Results}
\textbf{The null hypothesis is $C_\theta$ is a $t$ copula.}
\begin{enumerate}
    \item Use samples from Clayton copula generated by different simulation methods.

    \item Rejection rates are defined as the number of rejections of null hypothesis divided by the total number of repetitioins. Here, repeat each test 500 times to obtain the rejection rate.

    \item For RQMC, Halton sequence with a random start (left) and linear scrambled Faure sequences (right) are used.
\end{enumerate}
\begin{figure}[H]
\begin{minipage}[t]{0.49\linewidth}
\centering
\includegraphics[width=4in]{A1_Clt3.jpg}
\end{minipage}
\begin{minipage}[t]{0.49\linewidth}
\centering
\includegraphics[width=4in]{A1_Clt3_GS.jpg}
\end{minipage}
\end{figure}
\end{block}

\begin{block}{Conclusions/Current Research}	
\begin{enumerate}
    \item Better GoF tests: we designed a new test based on the collision test.

    \item Better simulation methods: we showed $GS^*$ improves rejection rates dramatically.
\end{enumerate}
\end{block}


\begin{block}{References}	
\small{\begin{thebibliography}{99}
\bibitem{overview}Berg, D. (2009). Copula goodness-of-fit testing: an overview and power comparison. \emph{The European Journal of Finance, 15(7-8), 675-701}.
\bibitem{modeldependence}Lindskog, F., RiskLab, E. T. H. (2000). Modelling dependence with copulas and applications to risk management. \emph{Doctoral dissertation, Master Thesis, ETH Z$\ddot{u}$rich}.
\bibitem{GS}Nguyen, N., $\ddot{\textup{O}}$kten, G. (2016). The acceptance-rejection method for low-discrepancy sequences. \emph{Monte Carlo Methods and Applications, 22(2), 133-148}.
\end{thebibliography}}
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