# A Review on Local Failure Probability Sensitivity Analysis
## 4 Local RSA in the Standard Normal Space
### Sensitivity through an Approximation of the Failure Domain Indicator Function
#### 4.3.2 Statistical Properties of the Weak Approach with Crude MC  
The statistical properties of this sensitivity estimate have been studied in numerous papers [20; 42; 50] as the choice of \(\widetilde{\sigma}\) is crucial and has influence on both the bias and the variance of the estimate. As the estimate is biased, it is interesting to use the expression of the mean square error denoted \(\widetilde{E}_{\text{mse}}^{\text{MC}}(\sigma)\), derived in [50], which is a function of \(\sigma>0\). \[\widetilde{E}_{\text{mse}}^{\text{MC}}(\sigma) =\left(\mathbb{E}\left[\frac{\partial\widetilde{P_{f}}(\mathbf{s},\sigma)}{\partial s_{\ell}}^{\text{MC}}\right]-\frac{\partial P_{f}(\mathbf{ s})}{\partial s_{\ell}}\right)^{2}+\mathbb{V}\mathrm{ar}\left(\frac{\partial \widetilde{P_{f}}(\mathbf{s},\sigma)}{\partial s_{\ell}}^{\text{MC}}\right)\] \[\widetilde{E}_{\text{mse}}^{\text{MC}}(\sigma) =e^{2}+\mathbb{V}\mathrm{ar}\left(\frac{\partial\widetilde{P_{f }}(\mathbf{s},\sigma)}{\partial s_{\ell}}^{\text{MC}}\right),\]  
where \(e\) is the bias. Intuitively, one would want the value of \(\widetilde{\sigma}\) to be as small as possible to reduce the bias. However, for very small \(\sigma\), the domain integral of \(\partial\widetilde{P_{f}}(\mathbf{s},\sigma)/\partial s_{\ell}\) converges to a surface integral over the limit state surface \(\{g(\mathbf{s},\mathbf{x})=0\}\). Consequently, most observations would then have a negligible weight in the mean sample value Equation (19) of the MC sensitivity estimate, as only few of them are close to the limit state surface. In [50], analytical formulas are derived when the uniform cdf defined on the interval \([-\sigma/2,-\sigma/2]\) is chosen as the approximation of the failure domain indicator function, for the one-dimensional case (\(d=1\)). The resulting bias \(e\) is proportional to \(\sigma^{2}\), thus \(e^{2}\propto\sigma^{4}\) and the variance is proportional to \(1/N\sigma\). The proportional property of the bias is extended to the multidimensional case in [20]. Consequently, decreasing the parameter \(\sigma\) greatly reduces the bias. Nevertheless, past a certain point, the variance of the estimate increases for smaller \(\sigma\) values, if \(N\) is kept constant. This is schematically illustrated in Figure 2. The horizontal axis is separated into two stages: the bias stage where the error is mainly affected by the bias of the estimate and the variance stage where the error is mainly affected by the variance of the estimate. Therefore, the optimal value of \(\widetilde{\sigma}\) should be selected at the intersection between the bias stage and the variance stage. This intersection value is challenging to obtain in practice and is problem dependent [51]. An analytical formula of the optimal \(\widetilde{\sigma}\) is detailed in [20] for the one-dimensional case, with the uniform cdf approximation mentioned above. This formula is proportional to \(1/N^{1/5}\) and depends on the quantity of interest \(\partial P_{f}(\mathbf{s})/\partial s_{\ell}\) as well as the derivative of order 3 of \(P_{f}\) with respect to \(s_{\ell}\); thus, its computation is intractable in practice. With this optimal \(\widetilde{\sigma}\) value, the value of the mean square error is computed and \(\widetilde{E}_{\text{mse}}^{\text{MC}}(\widetilde{\sigma})\propto 1/N^{4/5}\). This convergence rate is slower than the error convergence rate of the MC estimate obtained with the score function method, which is \(1/N\), see Section 3.2. Consequently, even with the best \(\widetilde{\sigma}\) value, the score function method is more efficient. However, the score function is applicable only when \(\mathbf{s}\) are distribution parameters. Finally, it should be underlined that the bias and variance of the sensitivity estimate are independent of \(P_{f}(\mathbf{s})\), as noticed in [50]. They are dependent on the derivatives of \(P_{f}\) of order 1 and above. In comparison, the mean square error of the MC estimate obtained with the score function method depends on \(P_{f}\), as shown in Equation (8).