Non-regular statistical estimation

统计学中的一些常见估计程序，如线性回归中的模型选择，可能对数据的微小变化或调整参数的变化极其敏感；这种程序可被认为是“非平稳的”，其统计性质通常非常复杂。例如，在考虑参数估计的不确定性时，应用统计学家往往忽视这种程序的敏感性或不稳定性，而标准统计方法通常低估了这种估计的不确定性。一些“非光滑”的估计过程(如模型选择)可以通过适当的正则化(例如在回归中使用套索)变得更加平滑，尽管这种正则化并不总是明显的。*本研究的一些目标可以概括如下：*(A)研究L无穷(切比雪夫)范数在统计建模中的使用，特别是从改善非参数函数估计行为的角度。特别地，我们将考虑最小化混合L无穷/L-2范数的过程的统计性质。*(B)研究线性回归模型中基本估计量的性质，以开发诊断程序并改进某些稳健估计量的计算和统计效率。我们还将研究随机抽样程序，以便在非常大的(即“大数据”)问题中近似最小二乘(和其他)估计**

Some common estimation procedures in statistics, such as model selection in linear regression, can be extremely sensitive to small changes in the data or to changes in tuning parameters; such procedures can be thought of "non-smooth" and their statistical properties are usually very complicated. For example, the sensitivity or instability of such procedures is often overlooked by applied statisticians when considering the uncertainty of parameter estimates and standard statistical methodology typically underestimates the uncertainty of such estimates. Some "non-smooth" estimation procedures (e.g. model selection) can be rendered more smooth by an appropriate regularization (e.g using the lasso in regression) although such regularization are not always obvious.***Some of the objectives of the proposed research can be summarized as follows: ***(a) Investigate the use of the L-infinity (Chebyshev) norm in statistical modeling, particularly from the perspective of improving the behaviour of non-parametric function estimation. In particular, we will consider the statistical properties of procedures that minimize a hybridized L-infinity/L-2 norm. ***(b) Investigate the properties of elemental estimators in linear regression models for the purposes of developing diagnostic procedures and improving the computational and statistical efficiency of certain robust estimators. We will also investigate random sampling procedures to approximate least squares (and other) estimation in very large (i.e. "big data") problems**