Converting the Survival Marshall–Olkin Copula to Its Ordinary Form

Summary

This note explains how to derive the ordinary Marshall–Olkin copula from its survival-copula form. It starts with two exponentially distributed lifetimes that may be affected by a shared failure event, then gives the survival copula in terms of parameters determined by the individual and common failure rates.

The conversion uses the standard relation between a bivariate survival copula and its ordinary copula: evaluate the survival copula at the complements of the ordinary uniform variables, then add the marginal probabilities and subtract one. The result is a formula for the ordinary copula when the inputs are the marginal cumulative probabilities. The document provides the transformation but does not expand it into a piecewise expression or discuss estimation, dependence measures, or applications to trading. Its scope is therefore a concise mathematical identity rather than an empirical analysis.

Key ideas

• The Marshall–Olkin survival copula is defined using marginal survival probabilities and parameters based on common and individual failure rates.
• The ordinary copula is obtained by evaluating the survival copula at the complements of its inputs.
• The conversion then adds the two ordinary marginal probabilities and subtracts one.
• The note states the relationship but does not explore estimation or practical applications.

Tags

# Relation of survival and non-survival Marshall-Olkin copula


# Relation of survival and non-survival Marshall-Olkin copula












Let us have two random variables $A$ and $B$ representing lifetimes of two elements of a system, where $A$ has cdf $F_A(x)$, $A \sim Exp(\lambda_1 + \lambda_{12})$ and $B$ has cdf $F_B(y)$, $B \sim Exp(\lambda_2 + \lambda_{12})$, with joint cdf $H(x,y)$ .

Marshall-Olkin copula is defined as survival copula $$\bar{H}(x,y) = C_{\theta_A, \theta_B}(u,v) = \min (u^{1-\theta_A}v, uv^{1-\theta_B})$$

where $$\theta_A = \frac{\lambda_{12}}{\lambda_1+\lambda_{12}} \text{ and } \theta_B = \frac{\lambda_{12}}{\lambda_2+\lambda_{12}}$$

$$u = \bar{F_A}(x) = 1 - F_A(x) \text{ and }v = \bar{F_B}(y) = 1- F_B(y)$$

What is the formula for non-survival Marshall-Olkin copula that would have $u = F_A(x)$ and $v=F_B(y)$?

## Answer by Gordon (score 2, accepted)

https://quant.stackexchange.com/a/26334

Note that the survival copula $C_{\theta_A, \theta_B}(u, v)$ and the non-survival copula $C(u, v)$ are related by \begin{align*} C_{\theta_A, \theta_B}(\hat{u}, \hat{v}) = \hat{u}+\hat{v}-1 + C(1-\hat{u}, 1-\hat{v}), \end{align*} where $\hat{u}=\bar{F}_A(x)=1 - F_A(x)$ and $\hat{v}=\bar{F}_B(y)=1-F_B(y)$. Then, \begin{align*} C(u, v) = C_{\theta_A, \theta_B}(1-u, 1-v) + u+v-1, \end{align*} where $u=F_A(x)$ and $v=F_B(y)$.

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