1,226,307 research outputs found
Limit distributions for large P\'{o}lya urns
We consider a two-color P\'{o}lya urn in the case when a fixed number of
balls is added at each step. Assume it is a large urn that is, the second
eigenvalue of the replacement matrix satisfies . After
drawings, the composition vector has asymptotically a first deterministic term
of order and a second random term of order . The object of
interest is the limit distribution of this random term. The method consists in
embedding the discrete-time urn in continuous time, getting a two-type
branching process. The dislocation equations associated with this process lead
to a system of two differential equations satisfied by the Fourier transforms
of the limit distributions. The resolution is carried out and it turns out that
the Fourier transforms are explicitly related to Abelian integrals over the
Fermat curve of degree . The limit laws appear to constitute a new family of
probability densities supported by the whole real line.Comment: Published in at http://dx.doi.org/10.1214/10-AAP696 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Computational Aspects of Optional P\'{o}lya Tree
Optional P\'{o}lya Tree (OPT) is a flexible non-parametric Bayesian model for
density estimation. Despite its merits, the computation for OPT inference is
challenging. In this paper we present time complexity analysis for OPT
inference and propose two algorithmic improvements. The first improvement,
named Limited-Lookahead Optional P\'{o}lya Tree (LL-OPT), aims at greatly
accelerate the computation for OPT inference. The second improvement modifies
the output of OPT or LL-OPT and produces a continuous piecewise linear density
estimate. We demonstrate the performance of these two improvements using
simulations
On the social opportunity cost of unemployment
The handling of unemployment is a central issue in cost-benefit analysis. Typically, the shadow price of employing an unemployed is derived by considering a marginal change in the employment constraint faced by an unemployed or rather an underemployed. In contrast, in this paper, we consider the discrete shift from unemployment to (full) employment. The result provides guidance how to estimate the social cost of recruiting otherwise unemployed to a project. It is shown that the social cost is overestimated by using the private reservation wage. The common practice of adding different cost items is shown to be flawed
On a preferential attachment and generalized P\'{o}lya's urn model
We study a general preferential attachment and Polya's urn model. At each
step a new vertex is introduced, which can be connected to at most one existing
vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is
not disconnected, it joins an existing pioneer vertex with probability
proportional to a function of the degree of that vertex. This function is
allowed to be vertex-dependent, and is called the reinforcement function. We
prove that there can be at most three phases in this model, depending on the
behavior of the reinforcement function. Consider the set whose elements are the
vertices with cardinality tending a.s. to infinity. We prove that this set
either is empty, or it has exactly one element, or it contains all the pioneer
vertices. Moreover, we describe the phase transition in the case where the
reinforcement function is the same for all vertices. Our results are general,
and in particular we are not assuming monotonicity of the reinforcement
function. Finally, consider the regime where exactly one vertex has a degree
diverging to infinity. We give a lower bound for the probability that a given
vertex ends up being the leading one, that is, its degree diverges to infinity.
Our proofs rely on a generalization of the Rubin construction given for
edge-reinforced random walks, and on a Brownian motion embedding.Comment: Published in at http://dx.doi.org/10.1214/12-AAP869 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …