Covariance Model Simulation Using Regular Vines

Abstract

We propose a new and flexible simulation method for non-normal data with user-specified marginal distributions, covariance matrix and certain bivariate dependencies. The VITA (VIne To Anything) method is based on regular vines and generalizes the NORTA (NORmal To Anything) method. Fundamental theoretical properties of the VITA method are deduced. Two illustrations demonstrate the flexibility and usefulness of VITA in the context of structural equation models. R code for the implementation is provided.

Appendix: Technical Results

Lemma 1

Let the edge e belong to an R-vine $\mathcal {V}$. Then $\mathcal {V}(e)$ is an R-vine.

Proof

The proximity condition is automatically satisfied, since each edge $e=\{a, b \}$ in $\mathcal {V}(e)$ also belongs to $\mathcal {V}$. Also, since the edges in $\mathcal {V}(e)$ also belongs to $\mathcal {V}$, there can be no cycles in $\mathcal {V}(e)$. Finally, let $e_1=\{a, b \}$ be an edge in $T_j(e)$. Then, by the proximity condition, $e_1$ corresponds to a sub-tree $T_1$, of $T_{j-1}(e)$ with two edges, a and b. Moreover, due to connectedness, $e_1$ shares a common node, say b, with some edge $e_2=\{b, c \}$ in $T_j(e)$, unless $T_j(e)$ is exhausted. Corresponding to $e_2$ is a sub-tree $T_2$ of $T_{j-1}(e)$ that shares edge b with $T_1$. Hence there is a path from a to c in $T_{j-1}(e)$.$\square$

Proof of Theorem 1

Let $e_k$ be the k’th edge in the order $\mathcal {O}$, and let $\mathfrak {c}_{e_k,a} = i, \mathfrak {c}_{e_k,b} = j$. We first show that $I_{e_k}(\theta _{e_k})$ is increasing. By Theorem 8.65 in Joe (2014), the CDF $F_{ij}$ of $(X_i,X_j)$ is increasing in $\theta _{e_k}$ since $\theta _{e_k} \mapsto B_{e_k, \theta _{e_k}}$ is increasing. Hence $I_{e_k}(\theta _{e_k})$ is increasing by the Hoeffding identity ${\text {Cov}} \, (X_i,X_j) = \int _{-\infty }^\infty \int _{-\infty }^\infty [ F_{i,j}(x,y) - F_i(x) F_j(x)] \, \mathrm{d}x \, \mathrm{d}y$. Theorem 8.65 omits the lowest tree $T_1$ from its statement, but here the result is immediate by the Hoeffding identity since $C_{i,j} = B_{e}$.

We now show continuity by generalizing Lemma 3 in Cario & Nelson (1997) (see also Lemma A.2 in Cario & Nelson, 1996): they use a stochastic representation specifically for the normal case, while ours is general. Let $\vartheta _{e_k} = (\theta _{e_1}, \theta _{e_2}, \ldots , \theta _{e_{k-1}})$, so that $(\theta _{e_1}, \theta _{e_2}, \ldots , \theta _{e_k}) = (\vartheta _{e_k}, \theta _{e_k})$. Note that $\vartheta _{e_k}$ is algebraically independent of $\theta _{e_k}$. Let $\delta = |U_{e_k}|$ be the cardinality of $U_{e_k}$. Suppose without loss of generality that $U_{e_k} = \{ 1, 2, \ldots , \delta \}$ and that $i = \delta -1$ and $j = \delta$. We now show continuity through showing that for any countable sequence $\{ \theta _{e_k, n} \}_{n \ge 1} \subseteq \Theta _{e_k}$ with $\lim _{n \rightarrow \infty } \theta _{e_k, n }= \theta _{e_k}$ we have $\lim _{n \rightarrow \infty } \mathbb {E} _{(\vartheta _{e_k}, \theta _{e_k,n})} X_{i} X_{j} = \mathbb {E} _{(\vartheta _{e_k}, \theta _{e_k})} X_{i} X_{j}$. We use the multivariate quantile transform of O’Brien (1975) (see Section 3 of Rüschendorf (2009) for a helpful discussion of this transformation). Let $F_{1,\ldots , \delta ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1}$ be the joint distribution induced by the parametric R-vine specification of $\mathcal {V}(e_k)$ with parameters $(\vartheta _{e_k}, \theta _{e_k})$, and let $F_{j| 1, \ldots , j-1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1}$ be the conditional quantile function of $F_{1,\ldots , \delta ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1}$ of the j’th variable, conditioned on the 1st through $j-1$’th variables.

Let $X(\vartheta _{e_k}, \theta _{e_k,n}) := (X_1(\vartheta _{e_k}, \theta _{e_k,n}), \ldots , X_\delta (\vartheta _{e_k}, \theta _{e_k,n}))'$ where the coordinates are defined recursively through $X_1(\vartheta _{e_k}, \theta _{e_k,n}) := F_{1, \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (U_1)$ and $X_{j}(\vartheta _{e,n}) := F_{j| 1, \ldots , j-1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (U_j | X_{1}(\vartheta _{e_k}, \theta _{e_k,n}), \ldots , X_{j-1}(\vartheta _{e_k}, \theta _{e_k,n}))$ for $2 \le j \le d$, where $U_1, U_2, \ldots , U_\delta \sim U[0,1]$ and independent. The multivariate quantile transform then gives $X(\vartheta _{e_k}, \theta _{e_k,n}) \sim F_{1, 2, \ldots , \delta ; \vartheta _{e_k}, \theta _{e_k,n}}$. Also let $X = ( X_1, \ldots , X_\delta )' := X (\vartheta _{e_k}, \theta _{e_k})$ so that $X \sim F_{1, \ldots , \delta ; \vartheta _{e_k}, \theta _{e_k}}$. We now show that for each $1 \le j \le d$ we have $\lim _{n \rightarrow \infty } X_j(\vartheta _{e_k}, \theta _{e_k,n}) = X_j(\vartheta _{e_k}, \theta _{e_k})$ a.s. (almost surely, i.e. with probability one), which implies that $\lim _{n \rightarrow \infty } X(\vartheta _{e_k}, \theta _{e_k,n}) = X (\vartheta _{e_k}, \theta _{e_k}) = X$ a.s. We proceed by induction. Firstly, $\theta _e \mapsto F_{1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (u_1)$ is assumed to be continuous. Hence, $\lim _{n \rightarrow \infty } X_1(\vartheta _{e,n}) = \lim _{n \rightarrow \infty } F_{1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (U_1) = F_{1 ; \vartheta _{e_k}, \theta _{e_k}}^ {-1} (U_1) = X_1$ a.s. by continuity. Let $2 \le j \le \delta$ and assume $\lim _{n \rightarrow \infty } X_i(\vartheta _{e_k}, \theta _{e_k,n}) = X_i$ a.s. for $1 \le i \le j-1$. By assumption, $(\theta _e, x_1, \ldots , x_{j-1}) \mapsto F_{j | 1, \ldots , j-1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (u_j | x_1, \ldots , x_{j-1} )$ is continuous. Hence, we complete the induction argument by concluding that $\lim _{n \rightarrow \infty } X_{j}(\vartheta _{e_k}, \theta _{e_k,n}) = \lim _{n \rightarrow \infty } F_{j| 1, \ldots , j-1 ; \vartheta _{e_k}, \theta _{e_k,n}}^ {-1} (U_j | X_{1}(\vartheta _{e,n}), \ldots , X_{j-1}(\vartheta _{e,n})) = X_j$ a.s. Let $\xrightarrow [n \rightarrow \infty ]{\mathscr {W}}$ denote convergence in distribution. The a.s. convergence implies $X(\vartheta _{e_k}, \theta _{e_k,n}) \xrightarrow [n \rightarrow \infty ]{\mathscr {W}} X(\vartheta _{e_k}, \theta _{e_k})$, implying $(X_{\delta -1}(\vartheta _{e_k}, \theta _{e_k,n}), X_{\delta }(\vartheta _{e_k}, \theta _{e_k,n}))' \xrightarrow [n \rightarrow \infty ]{\mathscr {W}} (X_{\delta -1}, X_\delta )'$.

The continuous mapping theorem (Billingsley, 1995, Theorem 29.2) implies that $X_{\delta -1}(\vartheta _{e_k}, \theta _{e_k,n}) X_{\delta }(\vartheta _{e_k}, \theta _{e_k,n}) \xrightarrow [n \rightarrow \infty ]{\mathscr {W}} X_{\delta -1} X_\delta$. Theorem 25.12 of Billingsley (1995) (i.e. uniform integrability and weak convergence implies moment convergence) and his eq. (25.13) (a moment condition which implies uniform integrability) shows that if $\sup _n \mathbb {E} |X_{\delta -1}(\vartheta _{e_k}, \theta _{e_k,n}) X_{\delta }(\vartheta _{e_k}, \theta _{e_k,n})|^{1 + \varepsilon } < \infty$, then we can transfer convergence in distribution to moment convergence, and we get $\lim _{n \rightarrow \infty } \mathbb {E} _{(\vartheta _{e_k}, \theta _{e_k,n})} X_{i} X_{j} = \mathbb {E} _{(\vartheta _{e_k}, \theta _{e_k})} X_{i} X_{j}$. And by the Cauchy–Schwarz inequality and the moment assumption, we have

$$\begin{aligned}&\sup _n \mathbb {E} |X_{\delta -1}(\vartheta _{e_k}, \theta _{e_k,n}) X_{\delta }(\vartheta _{e_k}, \theta _{e_k,n})|^{1 + \varepsilon /2} \\&\quad \le \sup _{\theta _{e_k} \in \Theta _{e_k}} \int _{[0,1]^d} |F_{i}^ {-1} (u_{i}) F_{j}^ {-1} (u_j)|^{1+\varepsilon /2} \, \mathrm{d}C_{i,j}(u ; \vartheta _{e_k}, \theta _{e_k}) \le \sup _{\theta \in \Theta } \mathbb {E} _\theta |X_{i} X_{j}|^{1+\varepsilon /2} \\&\quad \le \sup _{\theta \in \Theta } \sqrt{ \mathbb {E} _{\theta } |X_{i}|^{2+\varepsilon }} \sqrt{ \mathbb {E} _{\theta } |X_{j}|^{2+\varepsilon }} = \sqrt{\int _\mathbb {R} |x|^{2+\varepsilon } \, \mathrm{d}F_{i}(x) } \sqrt{ \int _\mathbb {R} |x|^{2+\varepsilon } \, \mathrm{d}F_{j}(x) } < \infty , \end{aligned}$$

where the equality follows since the copula parameterization does not change the marginals. This finishes the proof.$\square$