544 research outputs found
Fractional Operators, Dirichlet Averages, and Splines
Fractional differential and integral operators, Dirichlet averages, and
splines of complex order are three seemingly distinct mathematical subject
areas addressing different questions and employing different methodologies. It
is the purpose of this paper to show that there are deep and interesting
relationships between these three areas. First a brief introduction to
fractional differential and integral operators defined on Lizorkin spaces is
presented and some of their main properties exhibited. This particular approach
has the advantage that several definitions of fractional derivatives and
integrals coincide. We then introduce Dirichlet averages and extend their
definition to an infinite-dimensional setting that is needed to exhibit the
relationships to splines of complex order. Finally, we focus on splines of
complex order and, in particular, on cardinal B-splines of complex order. The
fundamental connections to fractional derivatives and integrals as well as
Dirichlet averages are presented
Exponential Splines of Complex Order
We extend the concept of exponential B-spline to complex orders. This
extension contains as special cases the class of exponential splines and also
the class of polynomial B-splines of complex order. We derive a time domain
representation of a complex exponential B-spline depending on a single
parameter and establish a connection to fractional differential operators
defined on Lizorkin spaces. Moreover, we prove that complex exponential splines
give rise to multiresolution analyses of and define wavelet
bases for
Bayesian nonparametric multivariate convex regression
In many applications, such as economics, operations research and
reinforcement learning, one often needs to estimate a multivariate regression
function f subject to a convexity constraint. For example, in sequential
decision processes the value of a state under optimal subsequent decisions may
be known to be convex or concave. We propose a new Bayesian nonparametric
multivariate approach based on characterizing the unknown regression function
as the max of a random collection of unknown hyperplanes. This specification
induces a prior with large support in a Kullback-Leibler sense on the space of
convex functions, while also leading to strong posterior consistency. Although
we assume that f is defined over R^p, we show that this model has a convergence
rate of log(n)^{-1} n^{-1/(d+2)} under the empirical L2 norm when f actually
maps a d dimensional linear subspace to R. We design an efficient reversible
jump MCMC algorithm for posterior computation and demonstrate the methods
through application to value function approximation
Estimating healthcare demand for an aging population: a flexible and robust bayesian joint model
In this paper, we analyse two frequently used measures of the demand for health care, namely hospital visits and out-of-pocket health care expenditure, which have been analysed separately in the existing literature. Given that these two measures of healthcare demand are highly likely to be closely correlated, we propose a framework to jointly model hospital visits and out-of-pocket medical expenditure. Furthermore, the joint framework allows for the presence of non-linear effects of covariates using splines to capture the effects of aging on healthcare demand. Sample heterogeneity is modelled robustly with the random effects following Dirichlet process priors with explicit cross-part correlation. The findings of our empirical analysis of the U.S. Health and Retirement Survey indicate that the demand for healthcare varies with age and gender and exhibits significant cross-part correlation that provides a rich understanding of how aging affects health care demand, which is of particular policy relevance in the context of an aging population
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