Optimum Thresholding Using Mean and Conditional Mean Square Error
Optimum Thresholding Using Mean and Conditional Mean Square Error
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使用均值和条件均方误差的最佳阈值
DOI：
10.2139/ssrn.3047108
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
C. Mancini
中科院分区：
文献类型：
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作者：
José E. Figueroa;C. Mancini
我们考虑一个资产价格（对数）的单变量半鞅模型，包含可能无限活动（IA）的跳跃。在[17]中提出的综合方差IV（s在ds中的sigma平方的积分，从0到T）的非参数阈值估计IV n是使用离散时间网格上的观测来构造的，并且精确地说，当它们低于阈值时，它对过程的平方增量求和，这是观测步长的确定性函数，并且可能是X的系数的确定性函数。所有满足给定条件的阈值函数都允许IV的渐近一致估计，然而hat（IV）的有限样本性质可能取决于阈值的具体选择。我们在这里的目标是最佳选择的阈值，最小化估计均方误差（MSE）或条件均方误差（cMSE）。最后一个标准允许达到一个阈值，这是最佳的，而不是在平均值，但对于特定的波动率和跳跃路径在手。一个简约的最佳特性的建立，这原来是渐近成比例的Lévy的连续性的基本布朗运动的模量。此外，最小化cMSE使我们能够提出一种新的实现方案，用于逼近最佳阈值。蒙特卡罗仿真表明了该方法的上级性能。
We consider a univariate semimartingale model for (the logarithm of) an asset price, containing jumps having possibly infinite activity (IA). The nonparametric threshold estimator IV n of the integrated variance IV (integral of sigma squared of s in ds, from 0 to T) proposed in [17] is constructed using observations on a discrete time grid, and precisely it sums up the squared increments of the process when they are below a threshold, a deterministic function of the observation step and possibly of the coefficients of X. All the threshold functions satisfying given conditions allow asymptotically consistent estimates of IV , however the finite sample properties of hat(IV) can depend on the specific choice of the threshold. We aim here at optimally selecting the threshold by minimizing either the estimation mean square error (MSE) or the conditional mean square error (cMSE). The last criterion allows to reach a threshold which is optimal not in mean but for the specific volatility and jumps paths at hand. A parsimonious characterization of the optimum is established, which turns out to be asymptotically proportional to the Lévy’s modulus of continuity of the underlying Brownian motion. Moreover, minimizing the cMSE enables us to propose a novel implementation scheme for approximating the optimal threshold. Monte Carlo simulations illustrate the superior performance of the proposed method.