Los Alamos National Laboratory
Publishes on Statistical Methods and Inference, Advanced Statistical Methods and Models, Bayesian Methods and Mixture Models. 10 papers and 3.4k citations.
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This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
Traditional response surface methodology focuses on modeling responses using parametric models with designs chosen to balance cost with adequate estimation of parameters and prediction in the design space. Using nonparametric smoothing to approximate the response surface offers both opportunities as well as problems. This article explores some conditions under which these methods can be appropriately used to increase the flexibility of surfaces modeled. The Box and Draper (1987) printing ink study is considered to illustrate the methods.
Summary Local polynomial regression is commonly used for estimating regression functions. In practice, however, with rough functions or sparse data, a poor choice of bandwidth can lead to unstable estimates of the function or its derivatives. We derive a new expression for the leading term of the bias by using the eigenvalues of the weighted design matrix where the bias depends on the arrangement of the X-values in the bandwidth window. We then use this result to determine a local data-driven bandwidth selection method and to provide a diagnostic for poor bandwidths that are chosen by using other methods. We show that our data-driven bandwidth is asymptotically equivalent to the optimal local bandwidth and that it performs well for relatively small samples when compared with other methods. In addition, we provide simulation results for first-derivative estimation. We illustrate its performance with data from Mars Global Surveyor.
Adaptive nonparametric kernel estimators for the location of a peak of the spectral density of a stationary time series are proposed and investigated. They are based on direct smoothing of the periodogram where the amount of smoothing is determined automatically in an asymptotically optimal fashion. These adaptive estimators minimize the asymptotic mean squared error. Adaptivity is derived from the weak convergence of a two-parameter stochastic process in a deviation and a bandwidth coordinate to a Gaussian limit process. Efficient global and local bandwidth choices which lead to adaptive peak estimators and practical aspects are discussed.