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Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. The Annals of Statistics 37
, 2009
"... We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequent ..."
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Cited by 7 (1 self)
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We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimaxoptimal up to a logarithmic factor, whatever the regularity level of the datagenerating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive. 1. Introduction. The
SHARP ADAPTIVE ESTIMATION OF THE DRIFT FUNCTION FOR ERGODIC DIFFUSIONS
, 2006
"... The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(·) is supposed to belong to a nonparametric class of smooth functions of order k ≥ 1, but the value of k is not known to the statistician. A fully datadriven proce ..."
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Cited by 7 (0 self)
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The global estimation problem of the drift function is considered for a large class of ergodic diffusion processes. The unknown drift S(·) is supposed to belong to a nonparametric class of smooth functions of order k ≥ 1, but the value of k is not known to the statistician. A fully datadriven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density. 1. Introduction. 1.1. The problem. In this paper we consider the statistical problem of estimating the drift function of a diffusion process X, given as the solution of the stochastic differential equation
Is Regularization Unnecessary for Boosting?
 Department of Statistics, Northwestern University
, 2001
"... this paper we present examples where `boosting forever ' leads to suboptimal predictions; while some regularization method, on the other hand, can achieve asymptotic optimality, at least in theory. We conjecture that this can be true in more general situations, and for some other regularization meth ..."
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Cited by 6 (0 self)
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this paper we present examples where `boosting forever ' leads to suboptimal predictions; while some regularization method, on the other hand, can achieve asymptotic optimality, at least in theory. We conjecture that this can be true in more general situations, and for some other regularization methods as well. Therefore the emerging literature on regularized variants of boosting is not unnecessary, but should be encouraged instead. The results of this paper are obtained from an analogy between some boosting algorithms that are used in regression and classification.
Tradeoffs Between Global And Local Risks In Nonparametric Function Estimation
"... We investigate the problem of loss adaptation: given a fixed parameter space we want to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. We consider the class of estimator sequences that achieve the minimax r ..."
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We investigate the problem of loss adaptation: given a fixed parameter space we want to construct an estimator that adapts to the loss function in the sense that the estimator is optimal both globally and locally at every point. We consider the class of estimator sequences that achieve the minimax rate, over a fixed Besov space, for estimating the entire function and establish a lower bound on the performance of any such sequence for estimation of the function at each point. This bound is larger by a logarithmic factor than the usual minimax rate for estimation at a point when the global and local minimax rates of convergence differ. We also consider estimators that achieve optimal minimax rates of convergence at every point and give a lower bound for the maximum global risk. An inequality concerning estimation in a two parameter statistical problem plays a key role in the proof. It can be considered as an generalization of an inequality in Brown and Low (1996b). This may be of independent interest. A particular wavelet estimator is constructed which is globally optimal and which attains the lower bound for the local risk provided by our inequality.
Adaptive Prediction and Estimation in Linear Regression With Infinitely Many Parameters
 Ann. Statist.,29
, 2001
"... The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in ` 2 . The method consists in an application of blockwise Stein's ..."
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The problem of adaptive prediction and estimation in the stochastic linear regression model with infinitely many parameters is considered. We suggest a prediction method that is sharp asymptotically minimax adaptive over ellipsoids in ` 2 . The method consists in an application of blockwise Stein's rule with "weakly" geometrically increasing blocks to the penalized least squares fits of the first N coefficients. To prove the results we develop oracle inequalities for sequence model with correlated data. Mathematics Subject Classifications: 62G05, 62G20 Key words: Linear regression with infinitely many parameters, adaptive prediction, exact asymptotics of minimax risk, blockwise Stein's rule, oracle inequalities. Short title: Adaptive prediction 1 Introduction Consider the regression model y = 1 X k=1 ` k x k + " (1) where fx k g k=1;2;::: is a sequence of explanatory variables, y is the corresponding response, " is the error, and ` = (` 1 ; ` 2 ; : : :) 2 ` 2 is an unknown regre...
On efficient estimation of linear functionals of a bivariate distribution with known marginals
 Statist. Probab. Lett
, 2002
"... In this paper we construct estimators for linear functionals of bivariate distributions with parametric marginals. Our construction generalizes the construction of efficient estimators given by Peng and Schick (2002) for the case of known marginals. More precisely, we show that in their constructi ..."
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In this paper we construct estimators for linear functionals of bivariate distributions with parametric marginals. Our construction generalizes the construction of efficient estimators given by Peng and Schick (2002) for the case of known marginals. More precisely, we show that in their construction the xed and known orthonormal bases for the Hilbert spaces of square integrable functions with zero means can now be replaced by estimated orthonormal bases using a n
Bargaining and Complex Preferences: Examining the Case of the Israeli Electorate
, 2001
"... on basic assumptions about individual preferences that are both empirically questionable and “not logically consistent with the basic assumptions on individual preferences made in economics.”(Milyo: 274) Two assumptions stand out in particular, that individuals have singlepeaked, strictly quasiconc ..."
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on basic assumptions about individual preferences that are both empirically questionable and “not logically consistent with the basic assumptions on individual preferences made in economics.”(Milyo: 274) Two assumptions stand out in particular, that individuals have singlepeaked, strictly quasiconcave preferences and that these are independent of other exogenous parameters. We argue in this paper that if voters do not have such wellbehaved preferences, intergroup bargains will be much harder to strike. Not only may optima be very difficult to locate but such equilibria are also more likely to be unstable. We then offer a new method to assess whether these assumptions hold empirically and apply it to the Israeli electorate. We find that in 1992 and 1996 the Israeli electorate clearly did not have Euclidean preferences and the evidence also strongly suggests that issues were not separable. The problem of peace in the Middle East has held a prominent place on the desk of American presidents for at least the last 25 years. After the Oslo Accords and especially in the last year of the Clinton Administration many were starting to become cautiously optimistic that an agreement between the Palestinians and the Israelis finally
NONPARAMETRIC CONFIDENCE SETS FOR DENSITIES Running Head: Nonparametric Confidence Sets
"... We present a method for constructing nonparametric confidence sets for density functions based on an approach due to Beran and Dümbgen (1998). We expand the density in an appropriate basis and we estimate the basis coefficients by using linear shrinkage methods. We then find the limiting distributio ..."
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We present a method for constructing nonparametric confidence sets for density functions based on an approach due to Beran and Dümbgen (1998). We expand the density in an appropriate basis and we estimate the basis coefficients by using linear shrinkage methods. We then find the limiting distribution of an asymptotic pivot based on the quadratic loss function. Inverting this pivot yields a confidence ball for the density. AMS 2000 subject classification. Primary 62G07, 62G15; secondary 62G20.
Adaptive Basis Density Estimation for HighDimensional Data
, 2010
"... All highdimensional density estimation techniques must make some assumptions about the underlying data distribution in order to be practical. In this proposal, I present work on a new method for high dimensional density estimation which assumes the ability to cheaply sample from an instrumental dis ..."
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All highdimensional density estimation techniques must make some assumptions about the underlying data distribution in order to be practical. In this proposal, I present work on a new method for high dimensional density estimation which assumes the ability to cheaply sample from an instrumental distribution which captures the lowdimensional structure in the data distribution. This assumption is satisfied in the application area of interest: modeling the distribution of tracks of tropical cyclones (TC) in the North Atlantic Ocean. Physical models are capable of generating realistic tracks, but not in the correct distribution over track space; my method allows for their use as instrumental distributions, anchoring the observed data in the vast highdimensional space. Using orthogonal series density estimation with a basis that is adapted to the instrumental distribution, I produce a density for the data distribution with respect not to the Lebesgue measure, but with respect to the instrumental distribution, which has the potential to improve the rates of convergence of quantities of interest. Initial simulations support this hypothesis. I propose to extend this work to conditional density estimation to allow for the introduction of covariates, which when applied to the TC track data will reveal the relationship between spatial locations of TCs and climatic predictors. Furthermore, I will explore plugin criteria for choosing optimal truncation points of the series, and for validating highdimensional density estimates. I will establish consistency results for the procedures. 1