Kurt S. Riedel

Kurt S. Riedel, A. Sidorenko

We determine the expected error by smoothing the data locally. Then we optimize the shape of the kernel smoother to minimize the error. Because the optimal estimator depends on the unknown function, our scheme automatically adjusts to the unknown function. By self-consistently adjusting the kernel smoother, the total estimator adapts to the data. Goodness of fit estimators select a kernel halfwidth by minimizing a function of the halfwidth which is based on the average square residual fit error: $ASR(h)$. A penalty term is included to adjust for using the same data to estimate the function and to evaluate the mean square error. Goodness of fit estimators are relatively simple to implement, but the minimum (of the goodness of fit functional) tends to be sensitive to small perturbations. To remedy this sensitivity problem, we fit the mean square error %goodness of fit functional to a two parameter model prior to determining the optimal halfwidth. Plug-in derivative estimators estimate the second derivative of the unknown function in an initial step, and then substitute this estimate into the asymptotic formula.

Alexander Sidorenko, Kurt S. Riedel

A hybrid estimator of the log-spectral density of a stationary time series is proposed. First, a multiple taper estimate is performed, followed by kernel smoothing the log-multitaper estimate. This procedure reduces the expected mean square error by $({π^2 \over 4})^{.8}$ over simply smoothing the log tapered periodogram. The optimal number of tapers is $O(N^{8/15})$. A data adaptive implementation of a variable bandwidth kernel smoother is given. When the spectral density is discontinuous, one sided smoothing estimates are used.

Kurt S. Riedel

Given noisy data, function estimation is considered when the unknown function is known apriori to consist of a small number of regions where the function is either convex or concave. When the regions are known apriori, the estimate is reduced to a finite dimensional convex optimization in the dual space. When the number of regions is unknown, the model selection problem is to determine the number of convexity change points. We use a pilot estimator based on the expected number of false inflection points.