87,367 research outputs found
Finite-Sample Maximum Likelihood Estimation of Location
We consider 1-dimensional location estimation, where we estimate a parameter
from samples , with each drawn i.i.d.
from a known distribution . For fixed the maximum-likelihood estimate
(MLE) is well-known to be optimal in the limit as : it is
asymptotically normal with variance matching the Cram\'er-Rao lower bound of
, where is the Fisher information of .
However, this bound does not hold for finite , or when varies with .
We show for arbitrary and that one can recover a similar theory based
on the Fisher information of a smoothed version of , where the smoothing
radius decays with .Comment: Corrected an inaccuracy in the description of the experimental setup.
Also updated funding acknowledgement
Efficient inference about the tail weight in multivariate Student distributions
We propose a new testing procedure about the tail weight parameter of
multivariate Student distributions by having recourse to the Le Cam
methodology. Our test is asymptotically as efficient as the classical
likelihood ratio test, but outperforms the latter by its flexibility and
simplicity: indeed, our approach allows to estimate the location and scatter
nuisance parameters by any root- consistent estimators, hereby avoiding
numerically complex maximum likelihood estimation. The finite-sample properties
of our test are analyzed in a Monte Carlo simulation study, and we apply our
method on a financial data set. We conclude the paper by indicating how to use
this framework for efficient point estimation.Comment: 23 page
A Robust Score-Driven Filter for Multivariate Time Series
A novel multivariate score-driven model is proposed to extract signals from
noisy vector processes. By assuming that the conditional location vector from a
multivariate Student's t distribution changes over time, we construct a robust
filter which is able to overcome several issues that naturally arise when
modeling heavy-tailed phenomena and, more in general, vectors of dependent
non-Gaussian time series. We derive conditions for stationarity and
invertibility and estimate the unknown parameters by maximum likelihood. Strong
consistency and asymptotic normality of the estimator are proved and the finite
sample properties are illustrated by a Monte-Carlo study. From a computational
point of view, analytical formulae are derived, which consent to develop
estimation procedures based on the Fisher scoring method. The theory is
supported by a novel empirical illustration that shows how the model can be
effectively applied to estimate consumer prices from home scanner data
Testing for Homogeneity in Mixture Models
Statistical models of unobserved heterogeneity are typically formalized as
mixtures of simple parametric models and interest naturally focuses on testing
for homogeneity versus general mixture alternatives. Many tests of this type
can be interpreted as tests, as in Neyman (1959), and shown to be
locally, asymptotically optimal. These tests will be contrasted
with a new approach to likelihood ratio testing for general mixture models. The
latter tests are based on estimation of general nonparametric mixing
distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator.
Recent developments in convex optimization have dramatically improved upon
earlier EM methods for computation of these estimators, and recent results on
the large sample behavior of likelihood ratios involving such estimators yield
a tractable form of asymptotic inference. Improvement in computation efficiency
also facilitates the use of a bootstrap methods to determine critical values
that are shown to work better than the asymptotic critical values in finite
samples. Consistency of the bootstrap procedure is also formally established.
We compare performance of the two approaches identifying circumstances in which
each is preferred
Efficient estimation of Banach parameters in semiparametric models
Consider a semiparametric model with a Euclidean parameter and an
infinite-dimensional parameter, to be called a Banach parameter. Assume: (a)
There exists an efficient estimator of the Euclidean parameter. (b) When the
value of the Euclidean parameter is known, there exists an estimator of the
Banach parameter, which depends on this value and is efficient within this
restricted model. Substituting the efficient estimator of the Euclidean
parameter for the value of this parameter in the estimator of the Banach
parameter, one obtains an efficient estimator of the Banach parameter for the
full semiparametric model with the Euclidean parameter unknown. This hereditary
property of efficiency completes estimation in semiparametric models in which
the Euclidean parameter has been estimated efficiently. Typically, estimation
of both the Euclidean and the Banach parameter is necessary in order to
describe the random phenomenon under study to a sufficient extent. Since
efficient estimators are asymptotically linear, the above substitution method
is a particular case of substituting asymptotically linear estimators of a
Euclidean parameter into estimators that are asymptotically linear themselves
and that depend on this Euclidean parameter. This more general substitution
case is studied for its own sake as well, and a hereditary property for
asymptotic linearity is proved.Comment: Published at http://dx.doi.org/10.1214/009053604000000913 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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