Non-regular statistical estimation

统计学中一些常见的估计过程，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；这样的程序可以被认为是“非光滑的”，它们的统计性质通常非常复杂。例如，应用统计学家在考虑参数估计的不确定性和标准统计方法通常低估了这种估计的不确定性时，往往忽略了这种程序的敏感性或不稳定性。一些“非平滑”的估计过程（例如模型选择）可以通过适当的正则化（例如在回归中使用套索）来呈现更平滑，尽管这种正则化并不总是明显的。***提出的一些研究目标可以概括如下：***(a)研究l -∞(Chebyshev)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，我们将考虑最小化杂化l -∞/L-2范数的过程的统计性质。***(b)研究线性回归模型中元素估计量的性质，以便发展诊断程序和提高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，以近似最小二乘（和其他）估计在非常大的(即。“大数据”)问题**

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**