97 research outputs found
Estimating linear functionals in nonlinear regression with responses missing at random
We consider regression models with parametric (linear or nonlinear)
regression function and allow responses to be ``missing at random.'' We assume
that the errors have mean zero and are independent of the covariates. In order
to estimate expectations of functions of covariate and response we use a fully
imputed estimator, namely an empirical estimator based on estimators of
conditional expectations given the covariate. We exploit the independence of
covariates and errors by writing the conditional expectations as unconditional
expectations, which can now be estimated by empirical plug-in estimators. The
mean zero constraint on the error distribution is exploited by adding suitable
residual-based weights. We prove that the estimator is efficient (in the sense
of H\'{a}jek and Le Cam) if an efficient estimator of the parameter is used.
Our results give rise to new efficient estimators of smooth transformations of
expectations. Estimation of the mean response is discussed as a special
(degenerate) case.Comment: Published in at http://dx.doi.org/10.1214/08-AOS642 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Inference about the slope in linear regression: an empirical likelihood approach
We present a new, efficient maximum empirical likelihood estimator for the slope in linear regression with independent errors and covariates. The estimator does not require estimation of the influence function, in contrast to other approaches, and is easy to obtain numerically. Our approach can also be used in the model with responses missing at random, for which we recommend a complete case analysis. This suffices thanks to results by Müller and Schick (Bernoulli 23:2693–2719, 2017), which demonstrate that efficiency is preserved. We provide confidence intervals and tests for the slope, based on the limiting Chi-square distribution of the empirical likelihood, and a uniform expansion for the empirical likelihood ratio. The article concludes with a small simulation study
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
Detecting heteroskedasticity in nonparametric regression using weighted empirical processes
Heteroskedastic errors can lead to inaccurate statistical conclusions if they are
not properly handled. We introduce a test for heteroskedasticity for the nonparametric regression
model with multiple covariates. It is based on a suitable residual-based empirical
distribution function. The residuals are constructed using local polynomial smoothing. Our
test statistic involves a "detection function" that can verify heteroskedasticity by exploiting
just the independence-dependence structure between the detection function and model
errors, i.e. we do not require a specific model of the variance function. The procedure is
asymptotically distribution free: inferences made from it do not depend on unknown parameters.
It is consistent at the parametric (root-n) rate of convergence. Our results are
extended to the case of missing responses and illustrated with simulations
The transfer principle: A tool for complete case analysis
This paper gives a general method for deriving limiting distributions of
complete case statistics for missing data models from corresponding results for
the model where all data are observed. This provides a convenient tool for
obtaining the asymptotic behavior of complete case versions of established full
data methods without lengthy proofs. The methodology is illustrated by
analyzing three inference procedures for partially linear regression models
with responses missing at random. We first show that complete case versions of
asymptotically efficient estimators of the slope parameter for the full model
are efficient, thereby solving the problem of constructing efficient estimators
of the slope parameter for this model. Second, we derive an asymptotically
distribution free test for fitting a normal distribution to the errors.
Finally, we obtain an asymptotically distribution free test for linearity, that
is, for testing that the nonparametric component of these models is a constant.
This test is new both when data are fully observed and when data are missing at
random.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1061 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
- …