*6.1. The Asymptotic Form of Contrast Q Indices for Small and Large Quantiles*

For small and large (design) quantiles, contrast *Q* indices can be rewritten using Equation (24) and the asymptotes of hyperbolic functions described in Chapter 3.3. The first-order contrast *Q* index can be rewritten as

$$Q\_l = \frac{l \cdot \alpha \cdot (1 - \alpha) - E((l|\mathbf{X}\_l) \cdot \alpha \cdot (1 - \alpha))}{l \cdot \alpha \cdot (1 - \alpha)} = \frac{l - E(l|\mathbf{X}\_l)}{l}.\tag{31}$$

By substituting the hyperbolic function *l* <sup>2</sup> <sup>−</sup> (θ\* <sup>−</sup> <sup>µ</sup>) <sup>2</sup> = σ 2 ·*l* 2 0 for *l*, we can obtain an approximate relation for *Q<sup>i</sup>* :

$$Q\_l \approx \frac{\sqrt{V(Y) \cdot l\_0 + (Q(Y) - E(Y))^2} - E\left(\sqrt{V(Y|X\_i) \cdot (l\_0|X\_i) + (Q(Y|X\_i) - E(Y|X\_i))^2}\right)}{\sqrt{V(Y) \cdot l\_0 + (Q(Y) - E(Y))^2}},\tag{32}$$

where *Q*(*Y*) = θ\*, *E*(*Y*) = µ*,* and *V*(*Y*) = σ 2 . The non-dimensional parameter *l*<sup>0</sup> can be calculated from Equation (25) as *l*<sup>0</sup> = *l* 2 /σ <sup>2</sup> at the point θ\* = µ. However, the precise value of *l*<sup>0</sup> is not important if |*Q*(*Y*)-*E*(*Y*)| is large and *l*<sup>0</sup> does not affect the asymptotes. By substituting the hyperbolic functions with their asymptotes, Equation (31) can be simplified as

$$Q\_i = \frac{l - E(l|\mathbf{X}\_i)}{l} \approx \frac{\left| Q(Y) - E(Y) \right| - E\left( \left| Q(\mathbf{Y}|\mathbf{X}\_i) - E(\mathbf{Y}|\mathbf{X}\_i) \right| \right)}{\left| Q(Y) - E(Y) \right|}. \tag{33}$$

Using asymptotes, the index is independent of variance and *l0*. The second-order probability *Q* index can be rewritten analogously:

$$Q\_{ij} \approx \frac{\left| Q(Y) - E(Y) \right| - E\left( \left| Q\left(Y \middle| \mathbf{X}\_{i\prime} \mathbf{X}\_{j} \right) - E\left(Y \middle| \mathbf{X}\_{i\prime} \mathbf{X}\_{j} \right) \right| \right)}{\left| Q(Y) - E(Y) \right|} - Q\_{i} - Q\_{j}. \tag{34}$$

The third-order probability *Q* index can be rewritten analogously:

$$Q\_{ijk} \approx \frac{\left| \mathbf{Q}(\mathbf{Y}) - \mathbf{E}(\mathbf{Y}) \right| - E\left( \left| \mathbf{Q}\{\mathbf{Y}|\mathbf{X}\_{i}, \mathbf{X}\_{j}, \mathbf{X}\_{k}\} - \mathbf{E}\{\mathbf{Y}|\mathbf{X}\_{i}, \mathbf{X}\_{j}, \mathbf{X}\_{k}\} \right| \right)}{\left| \mathbf{Q}(\mathbf{Y}) - \mathbf{E}(\mathbf{Y}) \right|} - Q\_{i} - Q\_{j} - Q\_{k} - Q\_{ij} - Q\_{ik} - Q\_{jk} \tag{35}$$

Equations (33)–(35) represent an asymptotic form of contrast *Q* indices that can be used for SAs of low and high quantiles. Higher-order contrast *Q* indices can be rewritten analogously. The sum of all indices thus estimated is equal to one.

In civil engineering, the design quantile of resistance tends to be less than the 0.01-quantile and the design quantile of resistance tends to be greater than the 0.99-quantile [4]. The asymptotic form of contrast *Q* indices reveals the degree of sensitivity as the distance between the quantile and the average value.

In the case study presented here, the use of Equation (33)–(35) leads to practically the same results as shown in Figure 14b, but only when low and high quantiles are analysed; otherwise, the formulas cannot be used. The computation of indices eliminates the repeated evaluation of contrast functions in the second loop.

*E*(*Y*|*Xi*))<sup>2</sup>

*b*.
