  Characterized by an outer integral connected to an inner integral through a
nonlinear function, nested integration is a challenging problem in various
fields, such as engineering and mathematical finance. The available numerical
methods for nested integration based on Monte Carlo (MC) methods can be
prohibitively expensive owing to the error propagating from the inner to the
outer integral. Attempts to enhance the efficiency of these approximations
using the quasi-MC (QMC) or randomized QMC (rQMC) method have focused on either
the inner or outer integral approximation. This work introduces a novel nested
rQMC method that simultaneously addresses the approximation of the inner and
outer integrals. The method leverages the unique nested integral structure to
offer a more efficient approximation mechanism. Incorporating Owen's scrambling
techniques, we address integrands exhibiting infinite variation in the
Hardy--Krause sense, enabling theoretically sound error estimates. As the
primary contribution, we derive asymptotic error bounds for the bias and
variance of our estimator, along with the regularity conditions under which
these bounds can be attained. Moreover, we derive a truncation scheme for
applications in the context of expected information gain estimation and
indicate how to use importance sampling to remedy the measure concentration
arising in the inner integral. We verify the estimator quality through
numerical experiments by comparing the computational efficiency of the nested
rQMC method against standard nested MC estimation for two case studies: one in
thermomechanics and the other in pharmacokinetics. These examples highlight the
computational savings and enhanced applicability of the proposed approach.
