100 research outputs found
Semiparametric inference in mixture models with predictive recursion marginal likelihood
Predictive recursion is an accurate and computationally efficient algorithm
for nonparametric estimation of mixing densities in mixture models. In
semiparametric mixture models, however, the algorithm fails to account for any
uncertainty in the additional unknown structural parameter. As an alternative
to existing profile likelihood methods, we treat predictive recursion as a
filter approximation to fitting a fully Bayes model, whereby an approximate
marginal likelihood of the structural parameter emerges and can be used for
inference. We call this the predictive recursion marginal likelihood.
Convergence properties of predictive recursion under model mis-specification
also lead to an attractive construction of this new procedure. We show
pointwise convergence of a normalized version of this marginal likelihood
function. Simulations compare the performance of this new marginal likelihood
approach that of existing profile likelihood methods as well as Dirichlet
process mixtures in density estimation. Mixed-effects models and an empirical
Bayes multiple testing application in time series analysis are also considered
Consistency of a recursive estimate of mixing distributions
Mixture models have received considerable attention recently and Newton
[Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for
estimating a mixing distribution. We prove almost sure consistency of this
recursive estimate in the weak topology under mild conditions on the family of
densities being mixed. This recursive estimate depends on the data ordering and
a permutation-invariant modification is proposed, which is an average of the
original over permutations of the data sequence. A Rao--Blackwell argument is
used to prove consistency in probability of this alternative estimate. Several
simulations are presented, comparing the finite-sample performance of the
recursive estimate and a Monte Carlo approximation to the permutation-invariant
alternative along with that of the nonparametric maximum likelihood estimate
and a nonparametric Bayes estimate.Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing
In the spirit of modeling inference for microarrays as multiple testing for
sparse mixtures, we present a similar approach to a simplified version of
quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the
number of tests usually reaches tens of thousands, the number of tests
performed in scans for QTL usually does not exceed several hundreds. However,
in typical cases, the sparsity of significant alternatives for QTL mapping
is in the same range as for microarrays. For methodological interest, as well
as some related applications, we also consider non-sparse mixtures. Using
simulations as well as theoretical observations we study false discovery rate
(FDR), power and misclassification probability for the Benjamini-Hochberg (BH)
procedure and its modifications, as well as for various parametric and
nonparametric Bayes and Parametric Empirical Bayes procedures. Our results
confirm the observation of Genovese and Wasserman (2002) that for small p the
misclassification error of BH is close to optimal in the sense of attaining the
Bayes oracle. This property is shared by some of the considered Bayes testing
rules, which in general perform better than BH for large or moderate 's.Comment: Published in at http://dx.doi.org/10.1214/193940307000000158 the IMS
Collections (http://www.imstat.org/publications/imscollections.htm) by the
Institute of Mathematical Statistics (http://www.imstat.org
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