Non-regular statistical estimation

统计学中的一些常见估计过程，例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非常敏感;这些过程可以被认为是“非平滑”的，并且它们的统计特性通常非常复杂。例如，应用统计学家在考虑参数估计值的不确定性时，往往会忽视这些程序的敏感性或不稳定性，而标准统计方法通常会低估这些估计值的不确定性。一些“非平滑”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）变得更加平滑，尽管这种正则化并不总是明显的。 拟议研究的一些目标可归纳如下： (a)研究在统计建模中使用L-infinity（Chebyshev）范数，特别是从改善非参数函数估计行为的角度。特别是，我们将考虑最小化混合L-无穷大/L-2范数的过程的统计特性。 (b)研究线性回归模型中元素估计量的性质，以开发诊断程序并提高某些稳健估计量的计算和统计效率。我们还将研究在非常大的（即“大数据”）问题中近似最小二乘（和其他）估计的随机抽样程序

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