## 1 Introduction
[PERSON]'s two-PEM (2 PEM) [14; 15] and [PERSON]'s three-PEM (3 PEM) [16; 17] are generally applied in the probabilistic framework for engineering problems, since they require only the first three and four order moments of the input uncertain variables, respectively. For more than three estimate points, moments higher than the fourth-order are necessary, which have less physical meaning in engineering and greatly increase computational burden [18]. However, the increased number of estimate points can improve the accuracy of PEM. In this regard, [PERSON] et al. proposed an improved three-PEM (I3 PEM) scheme for the probabilistic power flow calculation by adding a new pair of estimate points to [PERSON]'s three-PEM [18]. They concluded that the performance of I3 PEM is better than that of 3 PEM and 2 PEM in the results of power flow calculation with Normal distribution inputs, correlated inputs and bi-modal distribution inputs. Moreover, it avoided the calculation of higher order center moments. However, non-real number solutions may occur with normalized center distances since [PERSON]'s 3 PEM involves this in their method. Therefore, statistical moments of model output can be evaluated by the PEM, and the reconstruction of the PDF of model output becomes a moment problem. The moment problem indicates that the PDF of an uncertain variable is unique when knowing the infinite orders of moments and the non-uniqueness when less than three moments are known [19]. Apparently, knowing the infinite order moments of an uncertain variable is impossible, but the first four moments are very common in engineering since they all have clear physical meaning embedded in a large amount of probabilistic information. Therefore, plenty of parameterized probability distribution (PPD) methods for the first four moments have been proposed, such as saddlepoint approximation [13; 20], the maximum entropy model [21], the Pearson system and Johnson system [22] and shifted generalized lognormal distribution (SGLD) [23]. The core idea of PPD is to construct the unknown PDF by a parameterized function with undetermined parameters, which can be uniquely determined by the first four moments. As a result, the PEM combined with PPD can be an uncertainty propagation method for evaluating the unconditional and conditional PDF of seismic demand instead of the sampling-based method. The aforementioned discussions indicate that most of the reported works on global sensitivity analysis of seismic demand were conducted by tens of thousands of MC simulations. To address this limitation, an efficient moment-independent global sensitivity analysis is proposed in this paper. Inspired by [18], a modified four-point-estimated method (4 FEM) is derived to obtain the moments of a bridge's seismic demands by adding a new pair of points to the previous 2 PEM to avoid the non-real number solutions and the calculation of the fourth order moments of [PERSON]'s 3 PEM. Then, the unconditional and conditional probability density function (PDF) of seismic demands can be obtained by SGLD. The modified 4 PEM combined with the SGLD method are applied to MII analysis on the global sensitivity of the uncertain structural parameters on seismic demand for a bridge structure model. The ranking of the MII index can screen out which uncertain parameter has a significant influence on the seismic demand. The results are compared with those of the MC simulation method to verify their accuracy and the effectiveness of this method. The contents of this paper are divided as follows: In Section 2, the derivation of the proposed efficient MII method is addressed, in which the equations of the MII index are demonstrated in Section 2.1, a modified 4 PEM is derived in Section 2.2, the modified 4 PEM is combined with the SGLD method for MII analysis in Section 2.3, and Section 2.4 summarizes the procedures of the proposed method.