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Estimation of conditional quantiles by a new smoothing approximation of asymmetric loss functions

机译：通过不对称损失函数的新平滑近似来估计条件分位数

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摘要

In this paper, nonparametric estimation of conditional quantiles of a nonlinear time series model is formulated as a nonsmooth optimization problem involving an asymmetric loss function. This asymmetric loss function is nonsmooth and is of the same structure as the so-called 'lopsided' absolute value function. Using an effective smoothing approximation method introduced for this lopsided absolute value function, we obtain a sequence of approximate smooth optimization problems. Some important convergence properties of the approximation are established. Each of these smooth approximate optimization problems is solved by an optimization algorithm based on a sequential quadratic programming approximation with active set strategy. Within the framework of locally linear conditional quantiles, the proposed approach is compared with three other approaches, namely, an approach proposed by Yao and long (1996), the Iteratively Reweighted Least Squares method and the Interior-Point method, through some empirical numerical studies using simulated data and the classic lynx pelt series. In particular, the empirical performance of the proposed approach is almost identical with that of the Interior-Point method, both methods being slightly better than the Iteratively Reweighted Least Squares method. The Yao-Tong approach is comparable with the other methods in the ideal cases for the Yao-Tong method, but otherwise it is outperformed by other approaches. An important merit of the proposed approach is that it is conceptually simple and can be readily applied to parametrically nonlinear conditional quantile estimation.
机译：本文将非线性时间序列模型的条件分位数的非参数估计公式化为涉及不对称损失函数的非光滑优化问题。这种不对称损失函数是不平滑的，并且与所谓的“偏斜”绝对值函数具有相同的结构。使用针对此偏斜绝对值函数引入的有效平滑近似方法，我们获得了一系列近似平滑优化问题。建立了近似的一些重要收敛性质。这些平滑的近似优化问题中的每一个都是通过基于具有主动集策略的顺序二次规划逼近的优化算法来解决的。通过局部经验数值研究，在局部线性条件分位数的框架内，将该方法与其他三种方法进行了比较，即姚和朗（1996）提出的方法，迭代加权最小二乘方法和内部点方法。使用模拟数据和经典的山猫兽皮系列。特别是，该方法的经验性能几乎与内部点方法相同，这两种方法都比迭代加权最小二乘方法稍好。在理想的情况下，姚同方法与其他方法具有可比性，但在其他情况下却优于其他方法。所提出方法的一个重要优点是它在概念上很简单，可以很容易地应用于参数非线性条件分位数估计。

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