About

My research interests are in high dimensional integration, nonparametric confidence intervals, variable importance measures, and crossed random effects.

Research topics

• Mathematics
• Statistics
• Computer science
• Applied mathematics
• Combinatorics

Selected publications

• Randomized quasi-Monte Carlo for walk on spheres

  ArXiv.org · 2026-05-08

  articleOpen accessSenior author

  We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the integrands of interest are periodic indicator functions over regions $Θ$ in the torus $\mathbb{T}^k$. We give conditions for $\partialΘ$ to have $k-1$ dimensional Minkowski content which allows us to use results of He and Wang (2015). The RQMC estimates involve multiple values of $k$. We see sampling variances decreasing with the number $n$ of sample points at slightly better than Monte Carlo rates. The median variance rate in $4$ RQMC methods over $5$ worked examples, including some with $d=3$ and some with nonzero source functions, was slightly better than $O(n^{-1.1})$. The variance reduction factors ranged from $1.8$ to $10.7$ at $n=2^{17}$. None of the four RQMC methods dominated the others.

  PublisherOA PDF

• Quasi-Monte Carlo with one categorical variable

  Journal of Computational and Graphical Statistics · 2026-04-13

  articleOpen accessCorresponding

  We study randomized quasi-Monte Carlo (RQMC) estimation of a multivariate integral where one of the variables takes only a finite number of values. This problem arises when the variable of integration is drawn from a mixture distribution as is common in importance sampling and also arises in some recent work on transport maps. We find that when integration error decreases at an RQMC rate that it is then important to oversample the smallest mixture components instead of using a proportional allocation. This can even improve the rate of convergence. The optimal allocations depend on the possibly unknown convergence rate. Designing the sample with an incorrect assumption on the rate still attains that convergence rate, with an inferior implied constant. The penalty for using a pessimistic rate is typically higher than for using an optimistic one. We also find that for the most accurate RQMC sampling methods, it is advantageous to arrange that our $n=2^m$ randomized Sobol' points split into subsample sizes that are also powers of $2$.

  PublisherOA PDFDOI

• Walk on spheres and Array-RQMC

  ArXiv.org · 2026-05-13

  articleOpen accessSenior author

  We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al.\ (2025) that the mean dimension of Array-RQMC-WOS errors is much higher than an analogous Array-MC-WOS algorithm has.

  PublisherOA PDF

• Randomized quasi-Monte Carlo for walk on spheres

  arXiv (Cornell University) · 2026-05-08

  preprintOpen accessSenior author

  We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the integrands of interest are periodic indicator functions over regions $Θ$ in the torus $\mathbb{T}^k$. We give conditions for $\partialΘ$ to have $k-1$ dimensional Minkowski content which allows us to use results of He and Wang (2015). The RQMC estimates involve multiple values of $k$. We see sampling variances decreasing with the number $n$ of sample points at slightly better than Monte Carlo rates. The median variance rate in $4$ RQMC methods over $5$ worked examples, including some with $d=3$ and some with nonzero source functions, was slightly better than $O(n^{-1.1})$. The variance reduction factors ranged from $1.8$ to $10.7$ at $n=2^{17}$. None of the four RQMC methods dominated the others.

  PublisherDOI

• Walk on spheres and Array-RQMC

  arXiv (Cornell University) · 2026-05-13

  preprintOpen accessSenior author

  We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al.\ (2025) that the mean dimension of Array-RQMC-WOS errors is much higher than an analogous Array-MC-WOS algorithm has.

  PublisherDOI

• Error Estimation for Quasi-Monte Carlo

  Springer proceedings in mathematics & statistics · 2026-01-01 · 1 citations

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

• Empirical Bernstein and betting confidence intervals for randomized quasi-Monte Carlo

  ArXiv.org · 2025-04-25

  preprintOpen access

  Randomized quasi-Monte Carlo (RQMC) methods estimate the mean of a random variable by sampling an integrand at $n$ equidistributed points. For scrambled digital nets, the resulting variance is typically $\tilde O(n^{-θ})$ where $θ\in[1,3]$ depends on the smoothness of the integrand and $\tilde O$ neglects logarithmic factors. While RQMC can be far more accurate than plain Monte Carlo (MC) it remains difficult to get confidence intervals on RQMC estimates. We investigate some empirical Bernstein confidence intervals (EBCI) and hedged betting confidence intervals (HBCI), both from Waudby-Smith and Ramdas (2024), when the random variable of interest is subject to known bounds. When there are $N$ integrand evaluations partitioned into $R$ independent replicates of $n=N/R$ RQMC points, and the RQMC variance is $Θ(n^{-θ})$, then an oracle minimizing the width of a Bennett confidence interval would choose $n =Θ(N^{1/(θ+1)})$. The resulting intervals have a width that is $Θ(N^{-θ/(θ+1)})$. Our empirical investigations had optimal values of $n$ grow slowly with $N$, HBCI intervals that were usually narrower than the EBCI ones, and optimal values of $n$ for HBCI that were equal to or smaller than the ones for the oracle.

  PublisherOA PDFDOI

• Skewness of a randomized quasi-Monte Carlo estimate

  Journal of Complexity · 2025-05-05

  articleSenior authorCorresponding
  PublisherDOI

• Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)

  ArXiv.org · 2025-01-13

  preprintOpen access1st authorCorresponding

  Table 1 of Hall (1988) contains asymptotic coverage error formulas for some nonparametric approximate 95\% confidence intervals for the mean based on $n$ IID samples. The table includes an entry for an interval based on the central limit theorem using Gaussian quantiles and the Gaussian maximum likelihood variance estimate. It is missing an entry for the very widely used Student's $t$ confidence intervals. This note develops such a formula. The impetus to revisit this issue arose from the surprisingly robust performance of confidence intervals based on Student's t statistic in randomized quasi-Monte Carlo sampling. Hall's table had $0.14κ-2.12γ^2-3.35$ for normal theory intervals; the corresponding entry for Student's $t$ is $0.14κ-2.12γ^2$. An earlier version of this note reported that it corrected some coverage error formulas in Hall (1988). Two-sided errors take the form $2Φ^{-1}(0.975)(Aκ+ γ^2+C)φ(1.96)/n +O(1/n^{3/2})$ where the error may well be $O(n^{-2})$. Hall's table showed $Φ^{-1}(0.975)(Aκ+ Bγ^2+C)$. The version intended as a correction had $2(Aκ+ Bγ^2+C)$, wider by about $2/1.96\doteq1.02$. So, Hall's table really is proportional to the two-sided coverage errors.

  PublisherOA PDFDOI

• Scalable solutions for crossed random-effect models with random slopes

  Electronic Journal of Statistics · 2025-01-01

  articleOpen accessSenior author

  The crossed random effects model is widely used, finding applications in various fields such as longitudinal studies, e-commerce, and recommender systems, among others. However, these models encounter scalability challenges, as the computational time for standard algorithms grows superlinearly with the number N of observations in the data set, commonly O(N3∕2) or worse. Recent published works present scalable methods for crossed random effects in linear models and some generalized linear models, but those methods only allow for random intercepts. In this paper, we devise scalable algorithms for models that include random slopes. This addition brings substantial difficulty in estimating the random-effect covariance matrices in a scalable way. We address this issue by using a variational EM algorithm. Our proposed approach accommodates both diagonal covariance matrices and cases where no structure is assumed—a scenario common in fields such as psychology and neuroscience. In simulations, the proposed method is substantially faster than standard methods for large N. It is also more efficient than ordinary least squares which has a problem of greatly underestimating the sampling uncertainty in parameter estimates. We illustrate the new method on a MovieLens data set, as well as a large data set (five million observations) from the online retailer Stitch Fix.

  PublisherDOI

Recent grants

• Monte Carlo and Quasi-Monte Carlo Methods for Statistics

  NSF · $660k · 2009–2014

• Non-uniform sampling of permutations and large scale hypothesis testing

  NSF · $400k · 2015–2019

• BIGDATA: F: Computationally Efficient Algorithms for Large-Scale Crossed Random Effects Models

  NSF · $800k · 2018–2023

• Monte Carlo and Quasi-Monte Carlo Methods for Statistics

  NSF · $250k · 2006–2009

• Statistical Integration and Approximation

  NSF · $443k · 2003–2007

Frequent coauthors

• Adeline R. Whitney

  National Institutes of Health

  20 shared

• Stuart K. Kim

  Stanford Medicine

  17 shared

• Zexin Pan

  Stanford University

  17 shared

• Patrick O. Brown

  Stanford University

  16 shared

• Peter B. Jahrling

  16 shared

• Kathleen Rubins

  National Aeronautics and Space Administration

  16 shared

• John W. Huggins

  United States Army Medical Research Institute of Infectious Diseases

  16 shared

• Lisa E. Hensley

  Center for Grain and Animal Health Research

  16 shared

Awards & honors

• Elected to the National Academy of Sciences
• 2024 SIAM Fellow