25,485 research outputs found
A Greedy Algorithm for Unimodal Kernel Density Estimation by Data Sharpening
Nonparametric methods for smoothing, regression, and density estimation produce estimators with great shape flexibility. Although this flexibility is an advantage, the practical value of nonparametric methods would be increased if qualitative constraints—natural-language shape restrictions—could also be imposed on the estimator. In density estimation, the most common such constraints are monotonicity (the density must be nondecreasing or nonincreasing) and unimodality (the density must have only one peak). The work presented here takes unimodal kernel density estimation as a representative problem in constrained nonparametric estimation. The method proposed for handling the constraint is data sharpening. A greedy algorithm is described for achieving the unimodality constraint. The algorithm is deterministic and runs quickly. It can find solutions that are competitive with the incumbent method, sequential quadratic programming
Particle Learning for General Mixtures
This paper develops particle learning (PL) methods for the estimation of general mixture models. The approach is distinguished from alternative particle filtering methods in two major ways. First, each iteration begins by resampling particles according to posterior predictive probability, leading to a more efficient set for propagation. Second, each particle tracks only the "essential state vector" thus leading to reduced dimensional inference. In addition, we describe how the approach will apply to more general mixture models of current interest in the literature; it is hoped that this will inspire a greater number of researchers to adopt sequential Monte Carlo methods for fitting their sophisticated mixture based models. Finally, we show that PL leads to straight forward tools for marginal likelihood calculation and posterior cluster allocation.Business Administratio
Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions
A truncated sequential procedure is constructed for estimating the drift
coefficient at a given state point based on discrete data of ergodic diffusion
process. A nonasymptotic upper bound is obtained for a pointwise absolute error
risk. The optimal convergence rate and a sharp constant in the bounds are found
for the asymptotic pointwise minimax risk. As a consequence, the efficiency is
obtained of the proposed sequential procedure.Comment: Published at http://dx.doi.org/10.3150/14-BEJ655 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Sequential Model Selection Method for Nonparametric Autoregression
In this paper for the first time the nonparametric autoregression estimation
problem for the quadratic risks is considered. To this end we develop a new
adaptive sequential model selection method based on the efficient sequential
kernel estimators proposed by Arkoun and Pergamenshchikov (2016). Moreover, we
develop a new analytical tool for general regression models to obtain the non
asymptotic sharp or- acle inequalities for both usual quadratic and robust
quadratic risks. Then, we show that the constructed sequential model selection
proce- dure is optimal in the sense of oracle inequalities.Comment: 30 page
Sequential Quantiles via Hermite Series Density Estimation
Sequential quantile estimation refers to incorporating observations into
quantile estimates in an incremental fashion thus furnishing an online estimate
of one or more quantiles at any given point in time. Sequential quantile
estimation is also known as online quantile estimation. This area is relevant
to the analysis of data streams and to the one-pass analysis of massive data
sets. Applications include network traffic and latency analysis, real time
fraud detection and high frequency trading. We introduce new techniques for
online quantile estimation based on Hermite series estimators in the settings
of static quantile estimation and dynamic quantile estimation. In the static
quantile estimation setting we apply the existing Gauss-Hermite expansion in a
novel manner. In particular, we exploit the fact that Gauss-Hermite
coefficients can be updated in a sequential manner. To treat dynamic quantile
estimation we introduce a novel expansion with an exponentially weighted
estimator for the Gauss-Hermite coefficients which we term the Exponentially
Weighted Gauss-Hermite (EWGH) expansion. These algorithms go beyond existing
sequential quantile estimation algorithms in that they allow arbitrary
quantiles (as opposed to pre-specified quantiles) to be estimated at any point
in time. In doing so we provide a solution to online distribution function and
online quantile function estimation on data streams. In particular we derive an
analytical expression for the CDF and prove consistency results for the CDF
under certain conditions. In addition we analyse the associated quantile
estimator. Simulation studies and tests on real data reveal the Gauss-Hermite
based algorithms to be competitive with a leading existing algorithm.Comment: 43 pages, 9 figures. Improved version incorporating referee comments,
as appears in Electronic Journal of Statistic
Semiparametric theory for causal mediation analysis: Efficiency bounds, multiple robustness and sensitivity analysis
While estimation of the marginal (total) causal effect of a point exposure on
an outcome is arguably the most common objective of experimental and
observational studies in the health and social sciences, in recent years,
investigators have also become increasingly interested in mediation analysis.
Specifically, upon evaluating the total effect of the exposure, investigators
routinely wish to make inferences about the direct or indirect pathways of the
effect of the exposure, through a mediator variable or not, that occurs
subsequently to the exposure and prior to the outcome. Although powerful
semiparametric methodologies have been developed to analyze observational
studies that produce double robust and highly efficient estimates of the
marginal total causal effect, similar methods for mediation analysis are
currently lacking. Thus, this paper develops a general semiparametric framework
for obtaining inferences about so-called marginal natural direct and indirect
causal effects, while appropriately accounting for a large number of
pre-exposure confounding factors for the exposure and the mediator variables.
Our analytic framework is particularly appealing, because it gives new insights
on issues of efficiency and robustness in the context of mediation analysis. In
particular, we propose new multiply robust locally efficient estimators of the
marginal natural indirect and direct causal effects, and develop a novel double
robust sensitivity analysis framework for the assumption of ignorability of the
mediator variable.Comment: Published in at http://dx.doi.org/10.1214/12-AOS990 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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