7,340 research outputs found
Estimation of a Two-component Mixture Model with Applications to Multiple Testing
We consider a two-component mixture model with one known component. We
develop methods for estimating the mixing proportion and the unknown
distribution nonparametrically, given i.i.d.~data from the mixture model, using
ideas from shape restricted function estimation. We establish the consistency
of our estimators. We find the rate of convergence and asymptotic limit of the
estimator for the mixing proportion. Completely automated distribution-free
honest finite sample lower confidence bounds are developed for the mixing
proportion. Connection to the problem of multiple testing is discussed. The
identifiability of the model, and the estimation of the density of the unknown
distribution are also addressed. We compare the proposed estimators, which are
easily implementable, with some of the existing procedures through simulation
studies and analyse two data sets, one arising from an application in astronomy
and the other from a microarray experiment.Comment: 42 pages, 8 figures, 6 table
Estimation of constant and time-varying dynamic parameters of HIV infection in a nonlinear differential equation model
Modeling viral dynamics in HIV/AIDS studies has resulted in a deep
understanding of pathogenesis of HIV infection from which novel antiviral
treatment guidance and strategies have been derived. Viral dynamics models
based on nonlinear differential equations have been proposed and well developed
over the past few decades. However, it is quite challenging to use experimental
or clinical data to estimate the unknown parameters (both constant and
time-varying parameters) in complex nonlinear differential equation models.
Therefore, investigators usually fix some parameter values, from the literature
or by experience, to obtain only parameter estimates of interest from clinical
or experimental data. However, when such prior information is not available, it
is desirable to determine all the parameter estimates from data. In this paper
we intend to combine the newly developed approaches, a multi-stage
smoothing-based (MSSB) method and the spline-enhanced nonlinear least squares
(SNLS) approach, to estimate all HIV viral dynamic parameters in a nonlinear
differential equation model. In particular, to the best of our knowledge, this
is the first attempt to propose a comparatively thorough procedure, accounting
for both efficiency and accuracy, to rigorously estimate all key kinetic
parameters in a nonlinear differential equation model of HIV dynamics from
clinical data. These parameters include the proliferation rate and death rate
of uninfected HIV-targeted cells, the average number of virions produced by an
infected cell, and the infection rate which is related to the antiviral
treatment effect and is time-varying. To validate the estimation methods, we
verified the identifiability of the HIV viral dynamic model and performed
simulation studies.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS290 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error
This article considers estimation of constant and time-varying coefficients
in nonlinear ordinary differential equation (ODE) models where analytic
closed-form solutions are not available. The numerical solution-based nonlinear
least squares (NLS) estimator is investigated in this study. A numerical
algorithm such as the Runge--Kutta method is used to approximate the ODE
solution. The asymptotic properties are established for the proposed estimators
considering both numerical error and measurement error. The B-spline is used to
approximate the time-varying coefficients, and the corresponding asymptotic
theories in this case are investigated under the framework of the sieve
approach. Our results show that if the maximum step size of the -order
numerical algorithm goes to zero at a rate faster than , the
numerical error is negligible compared to the measurement error. This result
provides a theoretical guidance in selection of the step size for numerical
evaluations of ODEs. Moreover, we have shown that the numerical solution-based
NLS estimator and the sieve NLS estimator are strongly consistent. The sieve
estimator of constant parameters is asymptotically normal with the same
asymptotic co-variance as that of the case where the true ODE solution is
exactly known, while the estimator of the time-varying parameter has the
optimal convergence rate under some regularity conditions. The theoretical
results are also developed for the case when the step size of the ODE numerical
solver does not go to zero fast enough or the numerical error is comparable to
the measurement error. We illustrate our approach with both simulation studies
and clinical data on HIV viral dynamics.Comment: Published in at http://dx.doi.org/10.1214/09-AOS784 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On efficient estimators of the proportion of true null hypotheses in a multiple testing setup
We consider the problem of estimating the proportion of true null
hypotheses in a multiple testing context. The setup is classically modeled
through a semiparametric mixture with two components: a uniform distribution on
interval with prior probability and a nonparametric density
. We discuss asymptotic efficiency results and establish that two different
cases occur whether vanishes on a set with non null Lebesgue measure or
not. In the first case, we exhibit estimators converging at parametric rate,
compute the optimal asymptotic variance and conjecture that no estimator is
asymptotically efficient (i.e. attains the optimal asymptotic variance). In the
second case, we prove that the quadratic risk of any estimator does not
converge at parametric rate. We illustrate those results on simulated data
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