17,554 research outputs found
Nonlinear sequential designs for logistic item response theory models with applications to computerized adaptive tests
Computerized adaptive testing is becoming increasingly popular due to
advancement of modern computer technology. It differs from the conventional
standardized testing in that the selection of test items is tailored to
individual examinee's ability level. Arising from this selection strategy is a
nonlinear sequential design problem. We study, in this paper, the sequential
design problem in the context of the logistic item response theory models. We
show that the adaptive design obtained by maximizing the item information leads
to a consistent and asymptotically normal ability estimator in the case of the
Rasch model. Modifications to the maximum information approach are proposed for
the two- and three-parameter logistic models. Similar asymptotic properties are
established for the modified designs and the resulting estimator. Examples are
also given in the case of the two-parameter logistic model to show that without
such modifications, the maximum likelihood estimator of the ability parameter
may not be consistent.Comment: Published in at http://dx.doi.org/10.1214/08-AOS614 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Singular solutions for divergence-form elliptic equations involving regular variation theory: Existence and classification
We generalise and sharpen several recent results in the literature regarding
the existence and complete classification of the isolated singularities for a
broad class of nonlinear elliptic equations of the form \begin{equation} -{\rm
div}\,(\mathcal A(|x|) \,|\nabla u|^{p-2} \nabla u)+b(x)\,h(u)=0\quad \text{in
} B_1\setminus\{0\}, \end{equation} where denotes the open ball with
radius centred at zero in . We assume that
, and are positive functions associated with regularly varying functions
of index , and at , and respectively,
satisfying and . We prove that the
condition is sharp for the removability
of all singularities at zero for the positive solutions of our problem, where
denotes the "fundamental solution" of (the Dirac mass at zero) in , subject
to . If , we show that
any non-removable singularity at zero for a positive solution to our equation
is either weak (i.e., ) or strong
(). The main difficulty and novelty of
this paper, for which we develop new techniques, come from the explicit
asymptotic behaviour of the strong singularity solutions in the critical case,
which had previously remained open even for . We also study the
existence and uniqueness of the positive solution to our problem with a
prescribed admissible behaviour at zero and a Dirichlet condition on
Bootstrapping 2D CFTs in the Semiclassical Limit
We study two dimensional conformal field theories in the semiclassical limit.
In this limit, the four-point function is dominated by intermediate primaries
of particular weights along with their descendants, and the crossing equations
simplify drastically. For a four-point function receiving sufficiently small
contributions from the light primaries, the structure constants involving heavy
primaries follow a universal formula. Applying our results to the four-point
function of the twist field in the symmetric product orbifold, we
produce the Hellerman bound and the logarithmically corrected Cardy formula
that is valid for .Comment: 32 pages, 7 figures. v2, v3: references added, minor clarification
Carving Out the End of the World or (Superconformal Bootstrap in Six Dimensions)
We bootstrap superconformal field theories in six
dimensions, by analyzing the four-point function of flavor current multiplets.
Assuming flavor group, we present universal bounds on the central charge
and the flavor central charge . Based on the numerical data, we
conjecture that the rank-one E-string theory saturates the universal lower
bound on , and numerically determine the spectrum of long multiplets in
the rank-one E-string theory. We comment on the possibility of solving the
higher-rank E-string theories by bootstrap and thereby probing M-theory on
AdS/.Comment: 59 pages, 10 figures, 4 tables; v2-v5: typos corrected, references
adde
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