LS,LAD组合损失的高维统计性质分析

The statistical analysis of the combined loss of LS, LAD in high-dimension

主要针对损失函数为最小二乘LS(Least Squares)和最小绝对偏差LAD(Least Absolute Deviation)的凸组合形式,研究了观测数n和预测数p均趋于无穷大(lim n→∞p/n=κ,κ>0)时,高维稳健统计性质和高维罚稳健统计性质,得到了稳健估计和罚稳健估计的显示表达.结果显示这种凸组合损失函数的模型集成了LS和LAD损失的优点,同时消弱了它们的不足,具有优良的高维统计性质.

This article studies a convex combination of the Least Squares（LS） and Least Absolute Devia- tion（LAD）. By studying the robust statistical properties of high-dimensional and penalized robust statisticM properties of high dimension when the number of observations n and the number of prediction p tends to infinity （lim p/n=k, k 〉 0n-∞）, the expressions of robust estimation and penalized robust estimation are obtained. The result reveals that the loss function model of convex combination combines the advantages of the LS and LAD, at the same time, it relatively weakens their shortcomings, thus it has excellent high dimensional statistical properties.