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We derive Stein approximation bounds for functionals of uniform random variables, using chaos expansions and the Clark-Ocone representation formula combined with derivation and finite difference operators. This approach covers sums and…
We survey key techniques and results from approximation theory in the context of uniform approximations to real functions such as e^{-x}, 1/x, and x^k. We then present a selection of results demonstrating how such approximations can be used…
In this article we recover the distribution function (and possible density) of an arbitrary random variable that is subject to an additive measurement error. This problem is also known as deconvolution and has a long tradition in…
Efficient global optimization is the problem of minimizing an unknown function f, using as few evaluations f(x) as possible. It can be considered as a continuum-armed bandit problem, with noiseless data and simple regret. Expected…
We study adaptive approximation algorithms for general multivariate linear problems where the sets of input functions are non-convex cones. While it is known that adaptive algorithms perform essentially no better than non-adaptive…
The regret bound of an optimization algorithms is one of the basic criteria for evaluating the performance of the given algorithm. By inspecting the differences between the regret bounds of traditional algorithms and adaptive one, we…
We provide an upper bound as a random variable for the functions of estimators in high dimensions. This upper bound may help establish the rate of convergence of functions in high dimensions. The upper bound random variable may converge…
In certain applications involving the solution of a Bayesian inverse problem, it may not be possible or desirable to evaluate the full posterior, e.g. due to the high computational cost of doing so. This problem motivates the use of…
We investigate the use of conformal maps for the acceleration of convergence of the trapezoidal rule and Sinc numerical methods. The conformal map is a polynomial adjustment to the $\sinh$ map, and allows the treatment of a finite number of…
Many randomized approximation algorithms operate by giving a procedure for simulating a random variable $X$ which has mean $\mu$ equal to the target answer, and a relative standard deviation bounded above by a known constant $c$. Examples…
Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…
Symmetric elliptic integrals, which have been used as replacements for Legendre's integrals in recent integral tables and computer codes, are homogeneous functions of three or four variables. When some of the variables are much larger than…
In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…
Stochastic-approximation gradient methods are attractive for large-scale convex optimization because they offer inexpensive iterations. They are especially popular in data-fitting and machine-learning applications where the data arrives in…
We introduce a sampling framework to support approximate computing with estimated error bounds in Spark. Our framework allows sampling to be performed at the beginning of a sequence of multiple transformations ending in an aggregation…
A new nonparametric estimator of a convex regression function in any dimension is proposed and its convergence properties are studied. We start by using any estimator of the regression function and we \emph{convexify} it by taking the…
A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified…
Penalties that induce smoothness are common in nonparametric regression. In many settings, the amount of smoothness in the data generating function will not be known. Simon and Shojaie (2021) derived convergence rates for nonparametric…
Model approximations are common practice when estimating structural or quasi-structural models. The paper considers the econometric properties of estimators that utilize projections to reimpose information about the exact model in the form…
This paper provides inference methods for best linear approximations to functions which are known to lie within a band. It extends the partial identification literature by allowing the upper and lower functions defining the band to be any…