21,219 research outputs found
Quantile and Probability Curves Without Crossing
This paper proposes a method to address the longstanding problem of lack of
monotonicity in estimation of conditional and structural quantile functions,
also known as the quantile crossing problem. The method consists in sorting or
monotone rearranging the original estimated non-monotone curve into a monotone
rearranged curve. We show that the rearranged curve is closer to the true
quantile curve in finite samples than the original curve, establish a
functional delta method for rearrangement-related operators, and derive
functional limit theory for the entire rearranged curve and its functionals. We
also establish validity of the bootstrap for estimating the limit law of the
the entire rearranged curve and its functionals. Our limit results are generic
in that they apply to every estimator of a monotone econometric function,
provided that the estimator satisfies a functional central limit theorem and
the function satisfies some smoothness conditions. Consequently, our results
apply to estimation of other econometric functions with monotonicity
restrictions, such as demand, production, distribution, and structural
distribution functions. We illustrate the results with an application to
estimation of structural quantile functions using data on Vietnam veteran
status and earnings.Comment: 29 pages, 4 figure
Classification with unknown class-conditional label noise on non-compact feature spaces
We investigate the problem of classification in the presence of unknown
class-conditional label noise in which the labels observed by the learner have
been corrupted with some unknown class dependent probability. In order to
obtain finite sample rates, previous approaches to classification with unknown
class-conditional label noise have required that the regression function is
close to its extrema on sets of large measure. We shall consider this problem
in the setting of non-compact metric spaces, where the regression function need
not attain its extrema.
In this setting we determine the minimax optimal learning rates (up to
logarithmic factors). The rate displays interesting threshold behaviour: When
the regression function approaches its extrema at a sufficient rate, the
optimal learning rates are of the same order as those obtained in the
label-noise free setting. If the regression function approaches its extrema
more gradually then classification performance necessarily degrades. In
addition, we present an adaptive algorithm which attains these rates without
prior knowledge of either the distributional parameters or the local density.
This identifies for the first time a scenario in which finite sample rates are
achievable in the label noise setting, but they differ from the optimal rates
without label noise
Maximum L-likelihood estimation
In this paper, the maximum L-likelihood estimator (MLE), a new
parameter estimator based on nonextensive entropy [Kibernetika 3 (1967) 30--35]
is introduced. The properties of the MLE are studied via asymptotic analysis
and computer simulations. The behavior of the MLE is characterized by the
degree of distortion applied to the assumed model. When is properly
chosen for small and moderate sample sizes, the MLE can successfully trade
bias for precision, resulting in a substantial reduction of the mean squared
error. When the sample size is large and tends to 1, a necessary and
sufficient condition to ensure a proper asymptotic normality and efficiency of
MLE is established.Comment: Published in at http://dx.doi.org/10.1214/09-AOS687 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Cutset Sampling for Bayesian Networks
The paper presents a new sampling methodology for Bayesian networks that
samples only a subset of variables and applies exact inference to the rest.
Cutset sampling is a network structure-exploiting application of the
Rao-Blackwellisation principle to sampling in Bayesian networks. It improves
convergence by exploiting memory-based inference algorithms. It can also be
viewed as an anytime approximation of the exact cutset-conditioning algorithm
developed by Pearl. Cutset sampling can be implemented efficiently when the
sampled variables constitute a loop-cutset of the Bayesian network and, more
generally, when the induced width of the networks graph conditioned on the
observed sampled variables is bounded by a constant w. We demonstrate
empirically the benefit of this scheme on a range of benchmarks
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