Bandwidth selection for smooth backfitting in additive models

The smooth backfitting introduced by Mammen, Linton and Nielsen [Ann. Statist. 27 (1999) 1443-1490] is a promising technique to fit additive regression models and is known to achieve the oracle efficiency bound. In this paper, we propose and discuss three fully automated bandwidth selection methods for smooth backfitting in additive models. The first one is a penalized least squares approach which is based on higher-order stochastic expansions for the residual sums of squares of the smooth backfitting estimates. The other two are plug-in bandwidth selectors which rely on approximations of the average squared errors and whose utility is restricted to local linear fitting. The large sample properties of these bandwidth selection methods are given. Their finite sample properties are also compared through simulation experiments.Comment: Published at http://dx.doi.org/10.1214/009053605000000101 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A simple smooth backfitting method for additive models

In this paper a new smooth backfitting estimate is proposed for additive regression models. The estimate has the simple structure of Nadaraya--Watson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting. Each component is estimated with the same asymptotic accuracy as if the other components were known.Comment: Published at http://dx.doi.org/10.1214/009053606000000696 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Regular ternary triangular forms

An integer of the form $T_x=\frac{x(x+1)}2$ for some positive integer $x$ is called a triangular number. A ternary triangular form $aT_{x}+bT_{y}+cT_{z}$ for positive integers $a,b$ and $c$ is called regular if it represents every positive integer that is locally represented. In this article, we prove that there are exactly 49 primitive regular ternary triangular forms.Comment: 28 page

Color-Shift Measurement in Microlensing-Induced Stellar Variation from Future Space-Based Surveys

If a microlensing event is caused by a star, the event can exhibit change in color due to the light from the lens. In the previous and current lensing surveys, the color shift could not be used to constrain the lens population because the blended light responsible for the color shift is mostly attributed to nearby background stars rather than the lens. However, events to be observed in future space-based surveys do not suffer from blending and thus the color information can be used to constrain lenses. In this paper, we demonstrate the usefulness of future surveys in measuring color shifts. By conducting simulation of galactic lensing events based on the specification of a proposed space-based lensing survey, we estimate that the shift in the color of $R-H$ will be measured at 5$\sigma$ level for $\sim 12$ of events that occur on source stars with apparent magnitudes brighter than $J=22.5$. Color-shifted events tend to have high magnifications and the lenses will have brightnesses equivalent to those of source stars. The time scales of the color-shifted events tend to be longer than those without color shifts. From the mass distribution of lenses, we find that most of the color-shifted events will be produced by stellar lenses with spectral types down to mid M-type main sequence stars.Comment: 5 pages and 3 figure