89 research outputs found
Estimating invariant laws of linear processes by U-statistics
Suppose we observe an invertible linear process with independent mean-zero
innovations and with coefficients depending on a finite-dimensional parameter,
and we want to estimate the expectation of some function under the stationary
distribution of the process. The usual estimator would be the empirical
estimator. It can be improved using the fact that the innovations are centered.
We construct an even better estimator using the representation of the
observations as infinite-order moving averages of the innovations. Then the
expectation of the function under the stationary distribution can be written as
the expectation under the distribution of an infinite series in terms of the
innovations, and it can be estimated by a U-statistic of increasing order
(also called an ``infinite-order U-statistic'') in terms of the estimated
innovations. The estimator can be further improved using the fact that the
innovations are centered. This improved estimator is optimal if the
coefficients of the linear process are estimated optimally
Uniformly root- consistent density estimators for weakly dependent invertible linear processes
Convergence rates of kernel density estimators for stationary time series are
well studied. For invertible linear processes, we construct a new density
estimator that converges, in the supremum norm, at the better, parametric, rate
. Our estimator is a convolution of two different residual-based
kernel estimators. We obtain in particular convergence rates for such
residual-based kernel estimators; these results are of independent interest.Comment: Published at http://dx.doi.org/10.1214/009053606000001352 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Uniform convergence of convolution estimators for the response density in nonparametric regression
We consider a nonparametric regression model with a
random covariate that is independent of the error . Then the
density of the response is a convolution of the densities of
and . It can therefore be estimated by a convolution of kernel estimators
for these two densities, or more generally by a local von Mises statistic. If
the regression function has a nowhere vanishing derivative, then the
convolution estimator converges at a parametric rate. We show that the
convergence holds uniformly, and that the corresponding process obeys a
functional central limit theorem in the space of continuous
functions vanishing at infinity, endowed with the sup-norm. The estimator is
not efficient. We construct an additive correction that makes it efficient.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ451 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Efficient prediction for linear and nonlinear autoregressive models
Conditional expectations given past observations in stationary time series
are usually estimated directly by kernel estimators, or by plugging in kernel
estimators for transition densities. We show that, for linear and nonlinear
autoregressive models driven by independent innovations, appropriate smoothed
and weighted von Mises statistics of residuals estimate conditional
expectations at better parametric rates and are asymptotically efficient. The
proof is based on a uniform stochastic expansion for smoothed and weighted von
Mises processes of residuals. We consider, in particular, estimation of
conditional distribution functions and of conditional quantile functions.Comment: Published at http://dx.doi.org/10.1214/009053606000000812 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Optimality of estimators for misspecified semi-Markov models
Suppose we observe a geometrically ergodic semi-Markov process and have a
parametric model for the transition distribution of the embedded Markov chain,
for the conditional distribution of the inter-arrival times, or for both. The
first two models for the process are semiparametric, and the parameters can be
estimated by conditional maximum likelihood estimators. The third model for the
process is parametric, and the parameter can be estimated by an unconditional
maximum likelihood estimator. We determine heuristically the asymptotic
distributions of these estimators and show that they are asymptotically
efficient. If the parametric models are not correct, the (conditional) maximum
likelihood estimators estimate the parameter that maximizes the
Kullback--Leibler information. We show that they remain asymptotically
efficient in a nonparametric sense.Comment: To appear in a Special Volume of Stochastics: An International
Journal of Probability and Stochastic Processes
(http://www.informaworld.com/openurl?genre=journal%26issn=1744-2508) edited
by N.H. Bingham and I.V. Evstigneev which will be reprinted as Volume 57 of
the IMS Lecture Notes Monograph Series
(http://imstat.org/publications/lecnotes.htm
Asymptotically Optimal Estimation in Misspecified Time Series Models
A concept of asymptotically efficient estimation is presented whena misspecified parametric time series model is fitted to a stationary process.Efficiency of several minimum distance estimates is proved and the behavior ofthe Gaussian maximum likelihood estimate is studied. Furthermore, the behaviorof estimates that minimize the h-step prediction error is discussed briefly. The paper answers to some extent the question what happens when a misspecifiedmodel is fitted to time series data and one acts as if the model were true
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Non-standard behavior of density estimators for sums of squared observations
It has been shown recently that, under an appropriate integrability
condition, densities of functions of independent and identically
distributed random variables can be estimated at the parametric rate
by a local U-statistic, and a functional central limit theorem holds.
For the sum of two squared random variables, the integrability condition
typically fails. We show that then the estimator behaves differently
for different arguments.
At points in the support of the squared random variable, the rate
of the estimator slows down by a logarithmic factor and is independent
of the bandwidth, but the asymptotic variance depends on
the rate of the bandwidth, and otherwise only on the density of the
squared random variable at this point and at zero. A functional central
limit theorem cannot hold.
Of course, for bounded random variables, the sum of squares is
more spread out than a single square. At points outside the support of
the squared random variable, the estimator behaves classically. Now
the rate is again parametric, the asymptotic variance has a different
form and does not depend on the bandwidth, and a functional central
limit theorem holds
Estimation in Nonparametric Regression with Nonregular Errors
Abstract For sufficiently nonregular distributions with bounded support, the extreme observations converge to the boundary points at a faster rate than the square root of the sample size. In a nonparametric regression model with such a nonregular error distribution, this fact can be used to construct an estimator for the regression function that converges at a faster rate than the NadarayaWatson estimator. We explain this in the simplest case, review corresponding results from boundary estimation that are applicable here, and discuss possible improvements in parametric and semiparametric models
Pointwise convergence rates and central limit theorems for kernel density estimators in linear processes
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