# Distribution Approximation, Roth's Theorem, and Looking for Insects in Shipping Containers

Methods for distribution approximation, including the bootstrap, do not perform well when applied to lattice-valued data. For example, the inherent discreteness of lattice distributions confounds both the conventional normal approximation and the standard bootstrap when used to construct confidence intervals. However, in certain problems involving lattice-valued random variables, where more than one sample is involved, this difficulty can be overcome by ensuring that the ratios of sample sizes are quite irregular. For example, at least one of the ratios of sample sizes could be a reasonably good rational approximation to an irrational number. Results from number theory, in particular Roth's theorem (which applies to irrational numbers that are the roots of polynomials with rational coefficients), can be used to demonstrate theoretically the advantages of this approach. This project was motivated by a problem in risk analysis involving quarantine searches of shipping containers for insects and other environmental hazards, where confidence intervals for the sum of two binomial proportions are required.