Related papers: Walsh functions, scrambled $(0,m,s)$-nets, and neg…
We consider the problem of evaluating $I(\varphi):=\int_{[0,1)^s}\varphi(x) dx$ for a function $\varphi \in L^2[0,1)^{s}$. In situations where $I(\varphi)$ can be approximated by an estimate of the form $N^{-1}\sum_{n=0}^{N-1}\varphi(x^n)$,…
We introduce a new quality measure to assess randomized low-discrepancy point sets of finite size $n$. This new quality measure, which we call "pairwise sampling dependence index", is based on the concept of negative dependence. A negative…
Quasi-Monte Carlo (QMC) sampling has been developed for integration over $[0,1]^s$ where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error…
Classical algorithms in numerical analysis for numerical integration (quadrature/cubature) follow the principle of approximate and integrate: the integrand is approximated by a simple function (e.g. a polynomial), which is then integrated…
In a very recent work, Basu and Owen (2015) propose the use of scrambled geometric nets in numerical integration when the domain is a product of $s$ arbitrary spaces of dimension $d$ having a certain partitioning constraint. It was shown…
We study randomized quasi-Monte Carlo integration by scrambled nets. The scrambled net quadrature has long gained its popularity because it is an unbiased estimator of the true integral, allows for a practical error estimation, achieves a…
The $t$-value of a $(t, m, s)$-net is an important criterion of point sets for quasi-Monte Carlo integration, and many point sets are constructed in terms of the $t$-values, as this leads to small integration error bounds. Recently,…
We consider the problem of computing an approximation to the integral $I=\int_{[0,1]^d}f(x) dx$. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of $O(n^{-1/2})$ from $n$ independent random function evaluations.…
We propose a method to efficiently integrate truncated probability densities. The method uses Markov chain Monte Carlo method to sample from a probability density matching the function being integrated. The required normalisation or…
This paper proposes a family of weighted batch means variance estimators, which are computationally efficient and can be conveniently applied in practice. The focus is on Markov chain Monte Carlo simulations and estimation of the asymptotic…
Quasi-Monte Carlo (qMC) methods are a powerful alternative to classical Monte-Carlo (MC) integration. Under certain conditions, they can approximate the desired integral at a faster rate than the usual Central Limit Theorem, resulting in…
Importance sampling (IS) is valuable in reducing the variance of Monte Carlo sampling for many areas, including finance, rare event simulation, and Bayesian inference. It is natural and obvious to combine quasi-Monte Carlo (QMC) methods…
We introduce quasi-Monte Carlo rules for the numerical integration of functions $f$ defined on $[0,1]^s$, $s \ge 1$, which satisfy the following properties: the Fourier-, Fourier cosine- or Walsh coefficients of $f$ are absolutely summable…
We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\delta \ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces…
The computation of integrals is a fundamental task in the analysis of functional data, which are typically considered as random elements in a space of squared integrable functions. Borrowing ideas from recent advances in the Monte Carlo…
Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET)…
We motive and calculate Newton--Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that…
Motivated by recent work involving the analysis of leveraging spatial correlations in sparsified mean estimation, we present a novel procedure for constructing covariance estimator. The proposed Random-knots (Random-knots-Spatial) and…
In this paper, we analyse a method for approximating the distribution function and density of a random variable that depends in a non-trivial way on a possibly high number of independent random variables, each with support on the whole real…
We study numerical integration of functions $f: \mathbb{R}^{s} \to \mathbb{R}$ with respect to a probability measure. By applying the corresponding inverse cumulative distribution function, the problem is transformed into integrating an…