8,856 research outputs found
Quasi-likelihood for Spatial Point Processes
Fitting regression models for intensity functions of spatial point processes
is of great interest in ecological and epidemiological studies of association
between spatially referenced events and geographical or environmental
covariates. When Cox or cluster process models are used to accommodate
clustering not accounted for by the available covariates, likelihood based
inference becomes computationally cumbersome due to the complicated nature of
the likelihood function and the associated score function. It is therefore of
interest to consider alternative more easily computable estimating functions.
We derive the optimal estimating function in a class of first-order estimating
functions. The optimal estimating function depends on the solution of a certain
Fredholm integral equation which in practice is solved numerically. The
approximate solution is equivalent to a quasi-likelihood for binary spatial
data and we therefore use the term quasi-likelihood for our optimal estimating
function approach. We demonstrate in a simulation study and a data example that
our quasi-likelihood method for spatial point processes is both statistically
and computationally efficient
Estimating Components in Finite Mixtures and Hidden Markov Models
When the unobservable Markov chain in a hidden Markov model is stationary the marginal distribution of the observations is a finite mixture with the number of terms equal to the number of the states of the Markov chain. This suggests estimating the number of states of the unobservable Markov chain by determining the number of mixture components in the marginal distribution. We therefore present new methods for estimating the number of states in a hidden Markov model, and coincidentally the unknown number of components in a finite mixture, based on penalized quasi-likelihood and generalized quasi-likelihood ratio methods constructed from the marginal distribution. The procedures advocated are simple to calculate and results obtained in empirical applications indicate that they are as effective as current available methods based on the full likelihood. We show that, under fairly general regularity conditions, the methods proposed will generate strongly consistent estimates of the unknown number of states or components.Finite mixture, hidden Markov process, model selection, number of states, penalized quasi-likelihood, generalized quasi-likelihood ratio, strong consistency.
Quasi-Likelihood and/or Robust Estimation in High Dimensions
We consider the theory for the high-dimensional generalized linear model with
the Lasso. After a short review on theoretical results in literature, we
present an extension of the oracle results to the case of quasi-likelihood
loss. We prove bounds for the prediction error and -error. The results
are derived under fourth moment conditions on the error distribution. The case
of robust loss is also given. We moreover show that under an irrepresentable
condition, the -penalized quasi-likelihood estimator has no false
positives.Comment: Published in at http://dx.doi.org/10.1214/12-STS397 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
In high-dimensional model selection problems, penalized simple least-square
approaches have been extensively used. This paper addresses the question of
both robustness and efficiency of penalized model selection methods, and
proposes a data-driven weighted linear combination of convex loss functions,
together with weighted -penalty. It is completely data-adaptive and does
not require prior knowledge of the error distribution. The weighted
-penalty is used both to ensure the convexity of the penalty term and to
ameliorate the bias caused by the -penalty. In the setting with
dimensionality much larger than the sample size, we establish a strong oracle
property of the proposed method that possesses both the model selection
consistency and estimation efficiency for the true non-zero coefficients. As
specific examples, we introduce a robust method of composite L1-L2, and optimal
composite quantile method and evaluate their performance in both simulated and
real data examples
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