Non-regular statistical estimation

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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**