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Sep 23, 2013
09/13

by
Kengo Kato

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This paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function, and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate. Here we suppose that covariates are discretely observed and sampling points may differ across subjects, where the number of measurements per subject increases as the sample size. Also, we allow the quantile index to vary over a given subset...

Source: http://arxiv.org/abs/1202.4850v2

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49

Sep 21, 2013
09/13

by
Kengo Kato

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This paper aims at developing a quasi-Bayesian analysis of the nonparametric instrumental variables model, with a focus on the asymptotic properties of quasi-posterior distributions. In this paper, instead of assuming a distributional assumption on the data generating process, we consider a quasi-likelihood induced from the conditional moment restriction, and put priors on the function-valued parameter. We call the resulting posterior quasi-posterior, which corresponds to "Gibbs...

Source: http://arxiv.org/abs/1204.2108v4

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82

Sep 22, 2013
09/13

by
Kengo Kato

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This paper studies the statistical properties of the group Lasso estimator for high dimensional sparse quantile regression models where the number of explanatory variables (or the number of groups of explanatory variables) is possibly much larger than the sample size while the number of variables in "active" groups is sufficiently small. We establish a non-asymptotic bound on the $\ell_{2}$-estimation error of the estimator. This bound explains situations under which the group Lasso...

Source: http://arxiv.org/abs/1103.1458v2

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49

Sep 20, 2013
09/13

by
Kengo Kato

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This paper investigates the two-step estimation of a high dimensional additive regression model, in which the number of nonparametric additive components is potentially larger than the sample size but the number of significant additive components is sufficiently small. The approach investigated consists of two steps. The first step implements the variable selection, typically by the group Lasso, and the second step applies the penalized least squares estimation with Sobolev penalties to the...

Source: http://arxiv.org/abs/1207.5313v2

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9.0

Jun 29, 2018
06/18

by
Masaaki Imaizumi; Kengo Kato

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This paper studies a regression model where both predictor and response variables are random functions. We consider a functional linear model where the conditional mean of the response variable at each time point is given by a linear functional of the predictor variable. In this paper, we are interested in estimation of the integral kernel $b(s,t)$ of the conditional expectation operator, where $s$ is an output variable while $t$ is a variable that interacts with the predictor variable. This...

Topics: Statistics, Statistics Theory, Mathematics

Source: http://arxiv.org/abs/1609.00286

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6.0

Jun 30, 2018
06/18

by
Kengo Kato; Yuya Sasaki

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This paper develops a method to construct uniform confidence bands for a nonparametric regression function where a predictor variable is subject to a measurement error. We allow for the distribution of the measurement error to be unknown, but assume that there is an independent sample from the measurement error distribution. The sample from the measurement error distribution need not be independent from the sample on response and predictor variables. The availability of a sample from the...

Topics: Statistics Theory, Statistics, Mathematics

Source: http://arxiv.org/abs/1702.03377

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6.0

Jun 30, 2018
06/18

by
Antonio F. Galvao; Kengo Kato

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This paper considers fixed effects (FE) estimation for linear panel data models under possible model misspecification when both the number of individuals, $n$, and the number of time periods, $T$, are large. We first clarify the probability limit of the FE estimator and argue that this probability limit can be regarded as a pseudo-true parameter. We then establish the asymptotic distributional properties of the FE estimator around the pseudo-true parameter when $n$ and $T$ jointly go to...

Topics: Mathematics, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1403.2085

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20

Jun 26, 2018
06/18

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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We derive strong approximations to the supremum of the non-centered empirical process indexed by a possibly unbounded VC-type class of functions by the suprema of the Gaussian and bootstrap processes. The bounds of these approximations are non-asymptotic, which allows us to work with classes of functions whose complexity increases with the sample size. The construction of couplings is not of the Hungarian type and is instead based on the Slepian-Stein methods and Gaussian comparison...

Topics: Mathematics, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1502.00352

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50

Sep 21, 2013
09/13

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in probability theory, especially in empirical process and extreme value theories. Here we give explicit comparisons of expectations of smooth functions and distribution functions of maxima of Gaussian random vectors without any restriction on the covariance matrices. We also establish an anti-concentration...

Source: http://arxiv.org/abs/1301.4807v3

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183

Sep 23, 2013
09/13

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the supremum norm. We prove an abstract approximation theorem that is applicable to a wide variety of problems, primarily in statistics. In particular, the bound in the main approximation theorem is non-asymptotic and the theorem does not require uniform boundedness of the class of...

Source: http://arxiv.org/abs/1212.6885v4

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5.0

Jun 30, 2018
06/18

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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This paper derives central limit and bootstrap theorems for probabilities that sums of centered high-dimensional random vectors hit hyperrectangles and sparsely convex sets. Specifically, we derive Gaussian and bootstrap approximations for probabilities $\Pr(n^{-1/2}\sum_{i=1}^n X_i\in A)$ where $X_1,\dots,X_n$ are independent random vectors in $\mathbb{R}^p$ and $A$ is a hyperrectangle, or, more generally, a sparsely convex set, and show that the approximation error converges to zero even if...

Topics: Mathematics, Statistics Theory, Statistics

Source: http://arxiv.org/abs/1412.3661

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101

Sep 23, 2013
09/13

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the original vectors. The key innovation of this result is that it applies even when the dimension of random vectors (p) is large compared to the sample size (n); in fact, p can be much larger than n. We...

Source: http://arxiv.org/abs/1212.6906v3

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Sep 23, 2013
09/13

by
Victor Chernozhukov; Denis Chetverikov; Kengo Kato

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Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the Smirnov-Bickel-Rosenblatt (SBR) condition; see e.g. Gine and Nickl (2010). This condition requires existence of a limit distribution of an extreme value type for a supremum of a studentized empirical process (equivalently, for a supremum of a Gaussian process with an equivalent covariance kernel). The principal contribution of this paper is to remove the need for SBR condition....

Source: http://arxiv.org/abs/1303.7152v2

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99

Jul 20, 2013
07/13

by
Alexandre Belloni; Victor Chernozhukov; Kengo Kato

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We develop uniformly valid confidence regions for a regression coefficient in a high-dimensional sparse LAD (least absolute deviation or median) regression model. The setting is one where the number of regressors p could be large in comparison to the sample size n, but only s < < n of them are needed to accurately describe the regression function. Our new methods are based on the instrumental LAD regression estimator that assembles the optimal estimating equation from either post...

Source: http://arxiv.org/abs/1304.0282v2

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85

Sep 23, 2013
09/13

by
Alexandre Belloni; Xiaohong Chen; Victor Chernozhukov; Kengo Kato

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In this work we consider series estimators for the conditional mean in light of three new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue constant that controls the ratio between the $L^{\infty}$ and $L^{2}$-norms, and (iii) maximal inequalities with data-dependent bounds for processes whose entropy integrals diverge at some rate. These technical tools allow us to contribute to the series literature, specifically the...

Source: http://arxiv.org/abs/1212.0442v1