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 $B_r$ denotes the open ball with radius $r>0$ centred at zero in $\mathbb{R}^N$ $(N\geq 2)$. We assume that $\mathcal{A} \in C^1(0,1]$, $b\in C(\bar{B_1}\setminus\{0\})$ and $h\in C[0,\infty)$ are positive functions associated with regularly varying functions of index $\vartheta$, $\sigma$ and $q$ at $0$, $0$ and $\infty$ respectively, satisfying $q>p-1>0$ and $\vartheta-\sigma<p<N+\vartheta$. We prove that the condition $b(x) \,h(\Phi)\not \in L^1(B_{1/2})$ is sharp for the removability of all singularities at zero for the positive solutions of our problem, where $\Phi$ denotes the "fundamental solution" of $-{\rm div}\,(\mathcal A(|x|)\, |\nabla u|^{p-2} \nabla u)=\delta_0$ (the Dirac mass at zero) in $B_1$, subject to $\Phi|_{\partial B_1}=0$. If $b(x) \,h(\Phi)\in L^1(B_{1/2})$, we show that any non-removable singularity at zero for a positive solution to our equation is either weak (i.e., $\lim_{|x|\to 0} u(x)/\Phi(|x|)\in (0,\infty)$) or strong ($\lim_{|x|\to 0} u(x)/\Phi(|x|)=\infty$). 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 $\mathcal{A}=1$. 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 $\partial B_1$

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 $\mathbb Z_2$ twist field in the symmetric product orbifold, we produce the Hellerman bound and the logarithmically corrected Cardy formula that is valid for $h \geq c/12$.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 ${\cal N}=(1,0)$ superconformal field theories in six dimensions, by analyzing the four-point function of flavor current multiplets. Assuming $E_8$ flavor group, we present universal bounds on the central charge $C_T$ and the flavor central charge $C_J$. Based on the numerical data, we conjecture that the rank-one E-string theory saturates the universal lower bound on $C_J$, 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${}_7\times{\rm S}^4$/$\mathbb{Z}_2$.Comment: 59 pages, 10 figures, 4 tables; v2-v5: typos corrected, references adde