2,044 research outputs found
Generalized least squares can overcome the critical threshold in respondent-driven sampling
In order to sample marginalized and/or hard-to-reach populations,
respondent-driven sampling (RDS) and similar techniques reach their
participants via peer referral. Under a Markov model for RDS, previous research
has shown that if the typical participant refers too many contacts, then the
variance of common estimators does not decay like , where is the
sample size. This implies that confidence intervals will be far wider than
under a typical sampling design. Here we show that generalized least squares
(GLS) can effectively reduce the variance of RDS estimates. In particular, a
theoretical analysis indicates that the variance of the GLS estimator is
. We then derive two classes of feasible GLS estimators. The first
class is based upon a Degree Corrected Stochastic Blockmodel for the underlying
social network. The second class is based upon a rank-two model. It might be of
independent interest that in both model classes, the theoretical results show
that it is possible to estimate the spectral properties of the population
network from the sampled observations. Simulations on empirical social networks
show that the feasible GLS (fGLS) estimators can have drastically smaller error
and rarely increase the error. A diagnostic plot helps to identify where fGLS
will aid estimation. The fGLS estimators continue to outperform standard
estimators even when they are built from a misspecified model and when there is
preferential recruitment.Comment: Submitte
Spectral clustering and the high-dimensional stochastic blockmodel
Networks or graphs can easily represent a diverse set of data sources that
are characterized by interacting units or actors. Social networks, representing
people who communicate with each other, are one example. Communities or
clusters of highly connected actors form an essential feature in the structure
of several empirical networks. Spectral clustering is a popular and
computationally feasible method to discover these communities. The stochastic
blockmodel [Social Networks 5 (1983) 109--137] is a social network model with
well-defined communities; each node is a member of one community. For a network
generated from the Stochastic Blockmodel, we bound the number of nodes
"misclustered" by spectral clustering. The asymptotic results in this paper are
the first clustering results that allow the number of clusters in the model to
grow with the number of nodes, hence the name high-dimensional. In order to
study spectral clustering under the stochastic blockmodel, we first show that
under the more general latent space model, the eigenvectors of the normalized
graph Laplacian asymptotically converge to the eigenvectors of a "population"
normalized graph Laplacian. Aside from the implication for spectral clustering,
this provides insight into a graph visualization technique. Our method of
studying the eigenvectors of random matrices is original.Comment: Published in at http://dx.doi.org/10.1214/11-AOS887 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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