97JCGS03\P0251-------------------------------------------------
An Approach to Diagnosing Total Variation Convergence of MCMC Algorithms
S. P. Brooks, P. Dellaportas, and G. O. Roberts
We introduce a convergence diagnostic procedure for MCMC that
operates by estimating total variation distances for the
distribution of the algorithm after certain numbers of iterations.
The method has advantages over many existing methods in terms of
applicability, utility, and interpretability. It can be used to
assess convergence of both marginal and joint posterior densities,
and we show how it can be applied to the two most commonly used MCMC
samplers---the Gibbs Sampler and the Metropolis Hastings algorithm.
In some cases, the computational burden of this method may be large,
but we show how lower dimensional analogues of the full-dimensional
method are available at a lower computational cost. Illustrative
examples highlight the utility and interpretability of the proposed
diagnostic, but also highlight some of its limitations.
Key Words: Gibbs sampler; $L^1$ Distance; Markov chain Monte Carlo;
Metropolis Hastings.
97JCGS03\P0266---------------------------------------------------------
Spline Estimation of Discontinuous Regression Functions
Ja-Yong Koo
This article deals with regression function estimation when the
regression function is smooth at all but a finite number of points.
An important question is: How can one produce discontinuous output
without knowledge of the location of discontinuity points? Unlike
most commonly used smoothers that tend to blur discontinuity in the
data, we need to find a smoother that can detect such discontinuity.
In this article, linear splines are used to estimate discontinuous
regression functions. A procedure of knot-merging is introduced for
the estimation of regression functions near discontinuous points.
The basic idea is to use multiple knots for spline estimates. We
use an automatic procedure involving the least squares method,
stepwise knot addition, stepwise basis deletion, knot-merging, and
the Bayes information criterion to select the final model. The
proposed method can produce discontinuous outputs. Numerical
examples using both simulated and real data are given to illustrate
the performance of the proposed method.
Key Words: Basis selection; BIC; Change-point; Discontinuity; Knot-merging;
Multiple knots; Regression functions; Splines; Stepwise basis
deletion; Stepwise knot addition.
97JCGS03\P0285----------------------------------------------------------
A New Test for Outlier Detection From a Multivariate Mixture Distribution
Suojin Wang, Wayne A. Woodward, H. L. Gray, Stephen Wiechecki, &
Stephen R. Sain}
The problem of testing an outlier from a multivariate mixture
distribution of several populations has many important applications
in practice. One particular example is in monitoring worldwide
nuclear testing, where we wish to detect whether an observed seismic
event is possibly a nuclear explosion (an outlier) by comparing it
with the training samples from mining blasts and earthquakes. The
combined population of seismic events from mining blasts and
earthquakes can be viewed as a mixture distribution. The classical
likelihood ratio test appears to not be applicable in our problem,
and in spite of the importance of this problem, little progress has
been made in the literature. This article proposes a simple
modified likelihood ratio test that overcomes the difficulties in
the current problem. Bootstrap techniques are used to approximate
the distribution of the test statistic. The advantages of the new
test are demonstrated via simulation studies. Some new
computational findings are also reported.
Key Words: Bootstrap; EM algorithm; Likelihood ratio test; Monte Carlo
simulation; Nuclear testing.
97JCGS03\P0300---------------------------------------------------------
A Deterministic Method for Robust Estimation of Multivariate
Location and Shape
Wendy L. Poston, Edward J. Wegman, Carey E. Priebe, & Jeffrey L. Solka
The existence of outliers in a data set and how to deal with them is
an important problem in statistics. The minimum volume ellipsoid
(MVE) estimator is a robust estimator of location and covariate
structure; however its use has been limited because there are few
computationally attractive methods. Determining the MVE consists of
two parts---finding the subset of points to be used in the estimate
and finding the ellipsoid that covers this set. This article
addresses the first problem. Our method will also allow us to
compute the minimum covariance determinant (MCD) estimator. The
proposed method of subset selection is called the effective
independence distribution (EID) method, which chooses the subset by
minimizing determinants of matrices containing the data. This
method is deterministic, yielding reproducible estimates of location
and scatter for a given data set. The EID method of finding the MVE
is applied to several regression data sets where the true estimate
is known. Results show that the EID method, when applied to these
data sets, produces the subset of data more quickly than
conventional procedures and that there is less than 6\% relative
error in the estimates. We also give timing results illustrating
the feasibility of our method for larger data sets. For the case of
10,000 points in 10 dimensions, the compute time is under 25
minutes.
Key Words: Minimum covariance determinant; Minimum volume ellipsoid;
Outliers; Robust estimators; Subset selection.
97JCGS03\P0314----------------------------------------------------------
Case Influence Analysis in Bayesian Inference
Eric T. Bradlow and Alan M. Zaslavsky
We demonstrate how case influence analysis, commonly used in
regression can be applied to Bayesian hierarchical models. Draws
from the joint posterior distribution of parameters are importance
weighted to reflect the effect of deleting each observation in turn;
the ensuing changes in the posterior distribution of each parameter
are displayed graphically. The procedure is particularly useful
when drawing a sample from the posterior distribution requires
extensive calculations (as with a Markov Chain Monte Carlo sampler).
The structure of hierarchical models, and other models with local
dependence, makes the importance weights inexpensive to calculate
with little additional programming. Some new alternative weighting
schemes are described that extend the range of problems in which
reweighting can be used to assess influence. Applications to a
growth curve model and a complex hierarchical model for opinion data
are described. Our focus on case influence on parameters is
complementary to other work that measures influence by distances
between posterior or predictive distributions.
Key Words: Case deletion diagnostics; Hierarchical models; Importance
weighting; Random effects models.
97JCGS03\P0332----------------------------------------------------------
Jump Detection in Regression Surfaces
Peihua Qiu and Brian Yandell
We consider the problem of locating jumps in regression surfaces. A
jump detection algorithm is suggested based on local least squares
estimation. This method requires $0(Nk)$ computations, where $N$ is
the sample size and $k$ is the window width of the neighborhood.
This property makes it possible to handle large data sets. The
conditions imposed on the jump location curves, the jump surfaces,
and the noise are mild. We demonstrate this method in detail with
some numerical examples.
Key Words: Image processing; Jump detection criterion; Jump location curves;
Jump surfaces; Least squares plane; Modification procedure; Slopes;
Threshold value.
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