89,260 research outputs found
Asymptotic equivalence and adaptive estimation for robust nonparametric regression
Asymptotic equivalence theory developed in the literature so far are only for
bounded loss functions. This limits the potential applications of the theory
because many commonly used loss functions in statistical inference are
unbounded. In this paper we develop asymptotic equivalence results for robust
nonparametric regression with unbounded loss functions. The results imply that
all the Gaussian nonparametric regression procedures can be robustified in a
unified way. A key step in our equivalence argument is to bin the data and then
take the median of each bin. The asymptotic equivalence results have
significant practical implications. To illustrate the general principles of the
equivalence argument we consider two important nonparametric inference
problems: robust estimation of the regression function and the estimation of a
quadratic functional. In both cases easily implementable procedures are
constructed and are shown to enjoy simultaneously a high degree of robustness
and adaptivity. Other problems such as construction of confidence sets and
nonparametric hypothesis testing can be handled in a similar fashion.Comment: Published in at http://dx.doi.org/10.1214/08-AOS681 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Feasible Multivariate Nonparametric Estimation Using Weak Separability
One of the main practical problems of nonparametric regression estimation is the curse of dimensionality. The curse of dimensionality arises because nonparametric regression estimates are dependent variable averages local to the point at which the regression function is to be estimated. The number of observations `local' to the point of estimation decreases exponentially with the number of dimensions. The consequence is that the variance of unconstrained nonparametric regression estimators of multivariate regression functions is often so great that the unconstrained nonparametric regression estimates are of no practical use. In this paper I propose a new estimation method of weakly separable multivariate nonparametric regression functions. Weak separability is a weaker condition than required by other dimension--reduction techniques, although similar asymptotic variance reductions obtain. Indeed, weak separability is weaker than generalized additivity (see Hardle and Linton, 1996 and Horowitz, 1998). The proposed estimator is relatively easy to compute. Theoretical results in this paper include (i) a uniform law of large numbers for marginal integration estimators, (ii) a uniform law of large numbers for marginal summation estimators, (iii) a uniform law of large numbers for my new nonparametric regression estimator for weakly separable regression functions, (iv) both a uniform strong and weak law of large numbers for U-statistics, and (v) three central limit theorems for my nonparametric regression estimator for weakly separable regression functions.
The Interval Property in Multiple Testing of Pairwise Differences
The usual step-down and step-up multiple testing procedures most often lack
an important intuitive, practical, and theoretical property called the interval
property. In short, the interval property is simply that for an individual
hypothesis, among the several to be tested, the acceptance sections of relevant
statistics are intervals. Lack of the interval property is a serious
shortcoming. This shortcoming is demonstrated for testing various pairwise
comparisons in multinomial models, multivariate normal models and in
nonparametric models. Residual based stepwise multiple testing procedures that
do have the interval property are offered in all these cases.Comment: Published in at http://dx.doi.org/10.1214/11-STS372 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On nonparametric and semiparametric testing for multivariate linear time series
We formulate nonparametric and semiparametric hypothesis testing of
multivariate stationary linear time series in a unified fashion and propose new
test statistics based on estimators of the spectral density matrix. The
limiting distributions of these test statistics under null hypotheses are
always normal distributions, and they can be implemented easily for practical
use. If null hypotheses are false, as the sample size goes to infinity, they
diverge to infinity and consequently are consistent tests for any alternative.
The approach can be applied to various null hypotheses such as the independence
between the component series, the equality of the autocovariance functions or
the autocorrelation functions of the component series, the separability of the
covariance matrix function and the time reversibility. Furthermore, a null
hypothesis with a nonlinear constraint like the conditional independence
between the two series can be tested in the same way.Comment: Published in at http://dx.doi.org/10.1214/08-AOS610 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On deconvolution of distribution functions
The subject of this paper is the problem of nonparametric estimation of a
continuous distribution function from observations with measurement errors. We
study minimax complexity of this problem when unknown distribution has a
density belonging to the Sobolev class, and the error density is ordinary
smooth. We develop rate optimal estimators based on direct inversion of
empirical characteristic function. We also derive minimax affine estimators of
the distribution function which are given by an explicit convex optimization
problem. Adaptive versions of these estimators are proposed, and some numerical
results demonstrating good practical behavior of the developed procedures are
presented.Comment: Published in at http://dx.doi.org/10.1214/11-AOS907 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A nonparametric approach to the estimation of lengths and surface areas
The Minkowski content of a body represents
the boundary length (for ) or the surface area (for ) of . A
method for estimating is proposed. It relies on a nonparametric
estimator based on the information provided by a random sample (taken on a
rectangle containing ) in which we are able to identify whether every point
is inside or outside . Some theoretical properties concerning strong
consistency, -error and convergence rates are obtained. A practical
application to a problem of image analysis in cardiology is discussed in some
detail. A brief simulation study is provided.Comment: Published at http://dx.doi.org/10.1214/009053606000001532 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonparametric inference of doubly stochastic Poisson process data via the kernel method
Doubly stochastic Poisson processes, also known as the Cox processes,
frequently occur in various scientific fields. In this article, motivated
primarily by analyzing Cox process data in biophysics, we propose a
nonparametric kernel-based inference method. We conduct a detailed study,
including an asymptotic analysis, of the proposed method, and provide
guidelines for its practical use, introducing a fast and stable regression
method for bandwidth selection. We apply our method to real photon arrival data
from recent single-molecule biophysical experiments, investigating proteins'
conformational dynamics. Our result shows that conformational fluctuation is
widely present in protein systems, and that the fluctuation covers a broad
range of time scales, highlighting the dynamic and complex nature of proteins'
structure.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS352 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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