Nonparametrics and Functional Data Analysis

  • Smoothing of Distributions A fundamental theorem states that the empirical distribution F_n based on the iid sample X_1,…,X_n converges uniformly and almost surely to the common distribution F of the random variables. We have studies smoothed versions of F_n in the form of splines and their derived densities.

  • Bias Bounds The bias of polynomial curves fitted by least-squares can be described as an integral of the derivative of the true regression function against a kernel. Such an expression can easily be transformed into an upper bound for the absolute value of the bias, which are of the form of a product of a quantity that depends on the design and another one that involves the true regression function. One can use such expressions to derive crude bounds for the potential bias involved in a regression fit.