arXiv subjects
Publications and source records attributed to Alfred Galichon.
In this paper we relate the Equilibrium Assignment Problem (EAP), which is underlying in several economics models, to a system of nonlinear equations that we call the "nonlinear Bernstein-Schrödinger system", which is well-known in the linear case, but whose nonlinear extension does not seem to have been studied. We apply this connection to derive an existence result for the EAP, and an efficient computational method.
This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement-related operators, and derive functional limit theory for the entire rearranged curve and its functionals. We also establish validity of the bootstrap for estimating the limit law of the the entire rearranged curve and its functionals. Our limit results are generic in that they apply to every estimator of a monotone econometric function, provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions. Consequently, our results apply to estimation of other econometric functions with monotonicity restrictions, such as demand, production, distribution, and structural distribution functions. We illustrate the results with an application to estimation of structural quantile functions using data on Vietnam veteran status and earnings.
Given a convex bounded domain $Ω$ in ${\mathbb{R}}^{d}$ and an integer $N\geq 2$, we associate to any jointly $N$-monotone $(N-1)$-tuplet $(u_1, u_2,..., u_{N-1})$ of vector fields from $% Ω$ into $\mathbb{R}^{d}$, a Hamiltonian $H$ on ${\mathbb{R}}^{d} \times {\mathbb{R}}^{d} ... \times {\mathbb{R}}^{d}$, that is concave in the first variable, jointly convex in the last $(N-1)$ variables such that for almost all $% x\in Ω$, \hbox{$(u_1(x), u_2(x),..., u_{N-1}(x))= \nabla_{2,...,N} H(x,x,...,x)$. Moreover, $H$ is $N$-sub-antisymmetric, meaning that $\sum% \limits_{i=0}^{N-1}H(σ^{i}(\mathbf{x}))\leq 0$ for all $\mathbf{x}% =(x_{1},...,x_{N})\in Ω^{N}$, $σ$ being the cyclic permutation on ${\mathbb{R}}^{d}$ defined by $σ(x_{1},x_2,...,x_{N})=(x_{2},x_{3},...,x_{N},x_{1})$. Furthermore, $H$ is $N$% -antisymmetric in a sense to be defined below. This can be seen as an extension of a theorem of E. Krauss, which associates to any monotone operator, a concave-convex antisymmetric saddle function. We also give various variational characterizations of vector fields that are almost everywhere $N$-monotone, showing that they are dual to the class of measure preserving $N$-involutions on $Ω$.
This paper applies a regularization procedure called increasing rearrangement to monotonize Edgeworth and Cornish-Fisher expansions and any other related approximations of distribution and quantile functions of sample statistics. Besides satisfying the logical monotonicity, required of distribution and quantile functions, the procedure often delivers strikingly better approximations to the distribution and quantile functions of the sample mean than the original Edgeworth-Cornish-Fisher expansions.
In this paper, we focus on efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson [25], that efficiency is characterized by a comonotonicity condition. The goal of this paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate setting is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and we address it using techniques from convex duality and optimal transportation.
We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope.
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.
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 $1-α$, 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 $1-α$. We demonstrate the utility of the improved point and interval estimates with an age-height growth chart example.
A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.