Maximum simulated likelihood estimation of mixed multinomial logit models requires evaluation of a multidimensional integral. Quasi-Monte Carlo (QMC) methods such as Halton sequences and modified Latin hypercube sampling are workhorse methods for integral approximation. Earlier studies explored the potential of sparse grid quadrature (SGQ), but SGQ suffers from negative weights. As an alternative to QMC and SGQ, we looked into the recently developed designed quadrature (DQ) method. DQ requires fewer nodes to get the same level of accuracy as of QMC and SGQ, is as easy to implement, ensures positivity of weights, and can be created on any general polynomial space. We benchmarked DQ against QMC in a Monte Carlo and an empirical study. DQ outperformed QMC in all considered scenarios, is practice-ready and has potential to become the workhorse method for integral approximation.

中文翻译：

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设计正交以近似最大模拟似然估计中的积分

混合多项 logit 模型的最大模拟似然估计需要评估多维积分。准蒙特卡罗 (QMC) 方法（例如 Halton 序列和修正的拉丁超立方体采样）是积分逼近的主要方法。早期的研究探索了稀疏网格正交 (SGQ) 的潜力，但 SGQ 受到负权重的影响。作为 QMC 和 SGQ 的替代方案，我们研究了最近开发的设计正交 (DQ) 方法。DQ 需要更少的节点来获得与 QMC 和 SGQ 相同级别的准确度，同样易于实现，确保权重的正性，并且可以在任何一般多项式空间上创建。我们在蒙特卡罗和实证研究中对 DQ 与 QMC 进行了基准测试。DQ 在所有考虑的场景中都优于 QMC，