104 research outputs found
Nonparametric deconvolution problem for dependent sequences
We consider the nonparametric estimation of the density function of weakly
and strongly dependent processes with noisy observations. We show that in the
ordinary smooth case the optimal bandwidth choice can be influenced by long
range dependence, as opposite to the standard case, when no noise is present.
In particular, if the dependence is moderate the bandwidth, the rates of
mean-square convergence and, additionally, central limit theorem are the same
as in the i.i.d. case. If the dependence is strong enough, then the bandwidth
choice is influenced by the strength of dependence, which is different when
compared to the non-noisy case. Also, central limit theorem are influenced by
the strength of dependence. On the other hand, if the density is supersmooth,
then long range dependence has no effect at all on the optimal bandwidth
choice.Comment: Published in at http://dx.doi.org/10.1214/07-EJS154 the Electronic
Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Well-Posedness of Measurement Error Models for Self-Reported Data
It is widely admitted that the inverse problem of estimating the distribution of a latent variable X* from an observed sample of X, a contaminated measurement of X*, is ill-posed. This paper shows that a property of self-reporting errors, observed from validation studies, is that the probability of reporting the truth is nonzero conditional on the true values, and furthermore, this property implies that measurement error models for self-reporting data are in fact well-posed. We also illustrate that the classical measurement error models may in fact be conditionally well-posed given prior information on the distribution of the latent variable X*.
Improved rates for Wasserstein deconvolution with ordinary smooth error in dimension one
This paper deals with the estimation of a probability measure on the real
line from data observed with an additive noise. We are interested in rates of
convergence for the Wasserstein metric of order . The distribution of
the errors is assumed to be known and to belong to a class of supersmooth or
ordinary smooth distributions. We obtain in the univariate situation an
improved upper bound in the ordinary smooth case and less restrictive
conditions for the existing bound in the supersmooth one. In the ordinary
smooth case, a lower bound is also provided, and numerical experiments
illustrating the rates of convergence are presented
Nonparametric regression for dependent data in the errors-in-variables problem
We consider the nonparametric estimation of the regression functions for dependent data. Suppose that the covariates are observed with additive errors in the data and we employ nonparametric deconvolution kernel techniques to estimate the regression functions in this paper. We investigate how the strength of time dependence affects the asymptotic properties of the local constant and linear estimators. We treat both short-range dependent and long-range dependent linear processes in a unified way and demonstrate that the long-range dependence (LRD) of the covariates affects the asymptotic properties of the nonparametric estimators as well as the LRD of regression errors does.local polynomial regression, errors-in-variables, deconvolution, ordinary smooth case, supersmooth case, linear processes, long-range dependence
Gaussian Process Structural Equation Models with Latent Variables
In a variety of disciplines such as social sciences, psychology, medicine and
economics, the recorded data are considered to be noisy measurements of latent
variables connected by some causal structure. This corresponds to a family of
graphical models known as the structural equation model with latent variables.
While linear non-Gaussian variants have been well-studied, inference in
nonparametric structural equation models is still underdeveloped. We introduce
a sparse Gaussian process parameterization that defines a non-linear structure
connecting latent variables, unlike common formulations of Gaussian process
latent variable models. The sparse parameterization is given a full Bayesian
treatment without compromising Markov chain Monte Carlo efficiency. We compare
the stability of the sampling procedure and the predictive ability of the model
against the current practice.Comment: 12 pages, 6 figure
Testing for equality between two transformations of random variables
Consider two random variables contaminated by two unknown transformations.
The aim of this paper is to test the equality of those transformations. Two
cases are distinguished: first, the two random variables have known
distributions. Second, they are unknown but observed before contaminations. We
propose a nonparametric test statistic based on empirical cumulative
distribution functions. Monte Carlo studies are performed to analyze the level
and the power of the test. An illustration is presented through a real data
set.Comment: 15 page
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