Summary We consider a two‐component mixture model with one known component. We develop methods for estimating the mixing proportion and the unknown distribution non‐parametrically, given independent and identically distributed data from the mixture model, using ideas from shape‐restricted function estimation. We establish the consistency of our estimators. We find the rate of convergence and asymptotic limit of the estimator for the mixing proportion. Completely automated distribution‐free honest finite sample lower confidence bounds are developed for the mixing proportion. Connection to the problem of multiple testing is discussed. The identifiability of the model and the estimation of the density of the unknown distribution are also addressed. We compare the estimators proposed, which are easily implementable, with some of the existing procedures through simulation studies and analyse two data sets: one arising from an application in astronomy and the other from a microarray experiment.
We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernel methods, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the parametric and the nonparametric components given independent and identically distributed (i.i.d.) data. We prove the consistency and find the rates of convergence of the estimators. We establish n −1/2 -rate of convergence and the asymptotic efficiency of the parametric component under mild assumptions. A finite sample simulation corroborates our asymptotic theory and illustrates the superiority of our procedure over existing procedures. We also analyze a car mileage data set and a ozone concentration data set. The identifiability and existence of the PLSEs are also investigated.
Tidal debris structures formed from disrupted satellites contain important clues about the assembly histories of galaxies. To date, studies of these structures have been hampered by reliance on by-eye identification and morphological classification which leaves their interpretation significantly uncertain. In this work we present a new machinevision technique based on the Subspace-Constrained Mean Shift (SCMS) algorithm which can perform these tasks automatically. SCMS finds the location of the highdensity 'ridges' that define substructure morphology. After identification, the coefficients of an orthogonal series density estimator are used to classify points on the ridges as part of a continuum between shell-like or stream-like debris, from which a global morphological classification can be determined. We dub this procedure Subspace Constrained Unsupervised Detection of Structure (SCUDS). By applying this tool to controlled N-body simulations of minor mergers we demonstrate that the extracted classifications correspond to the well-understood underlying physics of phase mixing. The application of SCUDS to resolved stellar population data from near-future surveys will inform our understanding of the buildup of galaxies stellar halos.
In this paper we propose a new model-based smoothed bootstrap procedure for making inference on the maximum score estimator of Manski (1975Manski ( , 1985 and prove its consistency. We provide a set of sufficient conditions for the consistency of any bootstrap procedure in this problem. We compare the finite sample performance of different bootstrap procedures through simulations studies. The results indicate that our proposed smoothed bootstrap outperforms other bootstrap schemes, including the m-out-of-n bootstrap. Additionally, we prove a convergence theorem for triangular arrays of random variables arising from binary choice models, which may be of independent interest. JEL classification: C14; C25Keywords: Binary choice model, cube-root asymptotics, (in)-consistency of the bootstrap, latent variable model, smoothed bootstrap.
We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the errors have uniformly bounded conditional variance and have only finitely many moments. We show that the interplay between the moment assumptions on the error, the metric entropy of the class of functions involved, and the "local" structure of the function class around the truth drives the rate of convergence of the LSE. We find sufficient conditions on the errors under which the rate of the LSE matches the rate of the LSE under sub-Gaussian error. Our results are finite sample and allow for heteroscedastic and heavy-tailed errors.
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