A Smoothed-Distribution Form of Nadaraya-Watson Estimation

Given observation-pairs (xi ,yi ), i = 1,...,n , taken to be independent observations of the random pair (X ,Y), we sometimes want to form a nonparametric estimate of m(x) = E(Y/ X = x). Let YE have the empirical distribution of the yi , and let (XS ,YS ) have the kernel-smoothed distribution of the (xi ,yi ). Then the standard estimator, the Nadaraya-Watson form mNW(x) can be interpreted as E(YE?XS = x). The smoothed-distribution estimator ms (x)=E(YS/XS = x) is a more general form than  mNW (x) and often has better properties. Similar considerations apply to estimating Var(Y/X = x), and to local polynomial estimation. The discussion generalizes to vector (xi ,yi ).nonparametric regression, Nadaraya-Watson, kernel density, conditional expectation estimator, conditional variance estimator, local polynomial estimator

Commencement Exercises Program, August 10, 1945 - Transcription

Transcription of Bryant\u27s 82nd graduation ceremony. Includes remarks by Rhode Island Governor J. Howard McGrath, Pauline Bertha Fournier (student), Thomas J. Watson (International Business Machines Corporation President), Henry L. Jacobs (Bryant President), George A. Richards (Bryant faculty member), Milton Bolton (student) and Reverend Father Daniel M. Galliher

Harmonic measure for biased random walk in a supercritical Galton-Watson tree

We consider random walks $\lambda$-biased towards the root on a Galton-Watson tree, whose offspring distribution $(p_k)_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<\lambda<m$, the loop-erased trajectory of the biased random walk defines the $\lambda$-harmonic ray, whose law is the $\lambda$-harmonic measure on the boundary of the Galton-Watson tree. We answer a question of Lyons, Pemantle and Peres by showing that the $\lambda$-harmonic measure has a.s. strictly larger Hausdorff dimension than the visibility measure, which is the harmonic measure corresponding to the simple forward random walk. We also prove that the average number of children of the vertices along the $\lambda$-harmonic ray is a.s. bounded below by $m$ and bounded above by $m^{-1}\sum k^2 p_k$. Moreover, at least for $0<\lambda \leq 1$, the average number of children of the vertices along the $\lambda$-harmonic ray is a.s. strictly larger than that of the $\lambda$-biased random walk trajectory. We observe that the latter is not monotone in the bias parameter $\lambda$.Comment: revised version, accepted for publication in Bernoulli Journal. 18 pages, 1 figur