15,436 research outputs found
Information Geometry Approach to Parameter Estimation in Markov Chains
We consider the parameter estimation of Markov chain when the unknown
transition matrix belongs to an exponential family of transition matrices.
Then, we show that the sample mean of the generator of the exponential family
is an asymptotically efficient estimator. Further, we also define a curved
exponential family of transition matrices. Using a transition matrix version of
the Pythagorean theorem, we give an asymptotically efficient estimator for a
curved exponential family.Comment: Appendix D is adde
Semiparametric posterior limits
We review the Bayesian theory of semiparametric inference following Bickel
and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency
in parametric and semiparametric estimation problems, we consider the
Bernstein-von Mises theorem (see, e.g., Le Cam and Yang (1990)) and generalize
it to (LAN) regular and (LAE) irregular semiparametric estimation problems. We
formulate a version of the semiparametric Bernstein-von Mises theorem that does
not depend on least-favourable submodels, thus bypassing the most restrictive
condition in the presentation of Bickel and Kleijn (2012). The results are
applied to the (regular) estimation of the linear coefficient in partial linear
regression (with a Gaussian nuisance prior) and of the kernel bandwidth in a
model of normal location mixtures (with a Dirichlet nuisance prior), as well as
the (irregular) estimation of the boundary of the support of a monotone family
of densities (with a Gaussian nuisance prior).Comment: 47 pp., 1 figure, submitted for publication. arXiv admin note:
substantial text overlap with arXiv:1007.017
Goodness-of-fit testing and quadratic functional estimation from indirect observations
We consider the convolution model where i.i.d. random variables having
unknown density are observed with additive i.i.d. noise, independent of the
's. We assume that the density belongs to either a Sobolev class or a
class of supersmooth functions. The noise distribution is known and its
characteristic function decays either polynomially or exponentially
asymptotically. We consider the problem of goodness-of-fit testing in the
convolution model. We prove upper bounds for the risk of a test statistic
derived from a kernel estimator of the quadratic functional based on
indirect observations. When the unknown density is smoother enough than the
noise density, we prove that this estimator is consistent,
asymptotically normal and efficient (for the variance we compute). Otherwise,
we give nonparametric upper bounds for the risk of the same estimator. We give
an approach unifying the proof of nonparametric minimax lower bounds for both
problems. We establish them for Sobolev densities and for supersmooth densities
less smooth than exponential noise. In the two setups we obtain exact testing
constants associated with the asymptotic minimax rates.Comment: Published in at http://dx.doi.org/10.1214/009053607000000118 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Count data models with variance of unknown form: an application to a hedonic model of worker absenteeism
We examine an econometric model of counts of worker absences due to illness in a sluggishly adjusting hedonic labor market. We compare three estimators that parameterize the conditional variance?least squares, Poisson, and negative binomial pseudo maximum likelihood?to generalized least squares (GLS) using nonparametric estimates of the conditional variance. Our data support the hedonic absenteeism model. Semiparametric GLS coefficients are similar in sign, magnitude, and statistical significance to coefficients where the mean and variance of the errors are specified ex ante. In our data, coefficient estimates are sensitive to a regressor list but not to the econometric technique, including correcting for possible heteroskedasticity of unknown form.Publicad
Data-driven efficient score tests for deconvolution problems
We consider testing statistical hypotheses about densities of signals in
deconvolution models. A new approach to this problem is proposed. We
constructed score tests for the deconvolution with the known noise density and
efficient score tests for the case of unknown density. The tests are
incorporated with model selection rules to choose reasonable model dimensions
automatically by the data. Consistency of the tests is proved
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