80 research outputs found
Validity of heavy traffic steady-state approximations in generalized Jackson Networks
We consider a single class open queueing network, also known as a generalized
Jackson network (GJN). A classical result in heavy-traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a reflected Brownian motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network. In this paper we resolve this open
problem by proving that the re-scaled stationary distribution of the GJN
converges to the stationary distribution of the RBM, thus validating a
so-called ``interchange-of-limits'' for this class of networks. Our method of
proof involves a combination of Lyapunov function techniques, strong
approximations and tail probability bounds that yield tightness of the sequence
of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
The Hough transform estimator
This article pursues a statistical study of the Hough transform, the
celebrated computer vision algorithm used to detect the presence of lines in a
noisy image. We first study asymptotic properties of the Hough transform
estimator, whose objective is to find the line that ``best'' fits a set of
planar points. In particular, we establish strong consistency and rates of
convergence, and characterize the limiting distribution of the Hough transform
estimator. While the convergence rates are seen to be slower than those found
in some standard regression methods, the Hough transform estimator is shown to
be more robust as measured by its breakdown point. We next study the Hough
transform in the context of the problem of detecting multiple lines. This is
addressed via the framework of excess mass functionals and modality testing.
Throughout, several numerical examples help illustrate various properties of
the estimator. Relations between the Hough transform and more mainstream
statistical paradigms and methods are discussed as well.Comment: Published at http://dx.doi.org/10.1214/009053604000000760 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Recovering convex boundaries from blurred and noisy observations
We consider the problem of estimating convex boundaries from blurred and
noisy observations. In our model, the convolution of an intensity function
is observed with additive Gaussian white noise. The function is assumed to
have convex support whose boundary is to be recovered. Rather than directly
estimating the intensity function, we develop a procedure which is based on
estimating the support function of the set . This approach is closely
related to the method of geometric hyperplane probing, a well-known technique
in computer vision applications. We establish bounds that reveal how the
estimation accuracy depends on the ill-posedness of the convolution operator
and the behavior of the intensity function near the boundary.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Woodroofe's one-armed bandit problem revisited
We consider the one-armed bandit problem of Woodroofe [J. Amer. Statist.
Assoc. 74 (1979) 799--806], which involves sequential sampling from two
populations: one whose characteristics are known, and one which depends on an
unknown parameter and incorporates a covariate. The goal is to maximize
cumulative expected reward. We study this problem in a minimax setting, and
develop rate-optimal polices that involve suitable modifications of the myopic
rule. It is shown that the regret, as well as the rate of sampling from the
inferior population, can be finite or grow at various rates with the time
horizon of the problem, depending on "local" properties of the covariate
distribution. Proofs rely on martingale methods and information theoretic
arguments.Comment: Published in at http://dx.doi.org/10.1214/08-AAP589 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bandits with Dynamic Arm-acquisition Costs
We consider a bandit problem where at any time, the decision maker can add
new arms to her consideration set. A new arm is queried at a cost from an
"arm-reservoir" containing finitely many "arm-types," each characterized by a
distinct mean reward. The cost of query reflects in a diminishing probability
of the returned arm being optimal, unbeknown to the decision maker; this
feature encapsulates defining characteristics of a broad class of
operations-inspired online learning problems, e.g., those arising in markets
with churn, or those involving allocations subject to costly resource
acquisition. The decision maker's goal is to maximize her cumulative expected
payoffs over a sequence of n pulls, oblivious to the statistical properties as
well as types of the queried arms. We study two natural modes of endogeneity in
the reservoir distribution, and characterize a necessary condition for
achievability of sub-linear regret in the problem. We also discuss a
UCB-inspired adaptive algorithm that is long-run-average optimal whenever said
condition is satisfied, thereby establishing its tightness
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