Improving Point and Interval Estimates of Monotone Functions by Rearrangement

Suppose that a target function is monotonic, namely, weakly increasing, and an available original estimate of this target function is not weakly increasing. Rearrangements, univariate and multivariate, transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function. Furthermore, suppose an original simultaneous confidence interval, which covers the target function with probability at least $1-\alpha$, is defined by an upper and lower end-point functions that are not weakly increasing. Then the rearranged confidence interval, defined by the rearranged upper and lower end-point functions, is shorter in length in common norms than the original interval and also covers the target function with probability at least $1-\alpha$. We demonstrate the utility of the improved point and interval estimates with an age-height growth chart example.Comment: 24 pages, 4 figures, 3 table

Improving point and interval estimates of monotone functions by rearrangement

Suppose that a target function is monotonic, namely weakly increasing, and an original estimate of this target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm by using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. The improvement property of the rearrangement also extends to the construction of confidence bands for monotone functions. Let l and u be the lower and upper endpoint functions of a simultaneous confidence interval [l,u] that covers the true function with probability (1-a), then the rearranged confidence interval, defined by the rearranged lower and upper end-point functions, is shorter in length in common norms than the original interval and covers the true function with probability greater or equal to (1-a). We illustrate the results with a computational example and an empirical example dealing with age-height growth charts. Please note: This paper is a revised version of cemmap working Paper CWP09/07.

Knightian Auctions

We study single-good auctions in a setting where each player knows his own valuation only within a constant multiplicative factor \delta{} in (0,1), and the mechanism designer knows \delta. The classical notions of implementation in dominant strategies and implementation in undominated strategies are naturally extended to this setting, but their power is vastly different. On the negative side, we prove that no dominant-strategy mechanism can guarantee social welfare that is significantly better than that achievable by assigning the good to a random player. On the positive side, we provide tight upper and lower bounds for the fraction of the maximum social welfare achievable in undominated strategies, whether deterministically or probabilistically

Improving estimates of monotone functions by rearrangement

Suppose that a target function is monotonic, namely, weakly increasing, and an original estimate of the target function is available, which is not weakly increasing. Many common estimation methods used in statistics produce such estimates. We show that these estimates can always be improved with no harm using rearrangement techniques: The rearrangement methods, univariate and multivariate, transform the original estimate to a monotonic estimate, and the resulting estimate is closer to the true curve in common metrics than the original estimate. We illustrate the results with a computational example and an empirical example dealing with age-height growth charts.

Critical Behavior of the Random Potts Chain

We study the critical behavior of the random q-state Potts quantum chain by density matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains ($L \leq 16$) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar (Phys. Rev. Lett.{\bf 76}, 3001 (1996)). We show how an accurate analysis of moments of the distribution of magnetizations allows a precise determination of critical exponents, circumventing some problems related to binary disorder. Multiscaling properties of the model and dynamical correlation functions are also investigated.Comment: LaTeX2e file with Revtex, 9 pages, 8 eps figures, 4 tables; typos correcte

Monotonic growth of interlayer magnetoresistance in strong magnetic field in very anisotropic layered metals

It is shown, that the monotonic part of interlayer electronic conductivity strongly decreases in high magnetic field perpendicular to the conducting layers. We consider only the coherent interlayer tunnelling, and the obtained result strongly contradicts the standard theory. This effect appears in very anisotropic layered quasi-two-dimensional metals, when the interlayer transfer integral is less than the Landau level separation.Comment: 4 pages, no figure

Andreev Conductance of Chaotic and Integrable Quantum Dots

We examine the voltage V and magnetic field B dependent Andreev conductance of a chaotic quantum dot coupled via point contacts to a normal metal and a superconductor. In the case where the contact to the superconductor dominates, we find that the conductance is consistent with the dot itself behaving as a superconductor-- it appears as though Andreev reflections are occurring locally at the interface between the normal lead and the dot. This is contrasted against the behaviour of an integrable dot, where for a similar strong coupling to the superconductor, no such effect is seen. The voltage dependence of the Andreev conductance thus provides an extremely pronounced quantum signature of the nature of the dot's classical dynamics. For the chaotic dot, we also study non-monotonic re-entrance effects which occur in both V and B.Comment: 13 pages, 9 figure