Peter Hall, Australian National University
Qiwei Yao, London School of Economics
Abstract
Components of variance models arise in settings where the variation within a level has a relatively complex structure, in particular not being plausibly modellable as independent and identically distributed perturbations of a deterministic mean. In such cases, if each group effect can be replicated a large number of times, then standard methods can be used to estimate the distributions of both the group effect and the error. This cannot be achieved without replication, however. How feasible is distribution estimation if it is not possible to replicate prolifically? Can the distributions of random effects and errors be estimated consistently from a small number of replications of each of a large number of noisy group effects, for example in a nonparametric setting? Often extensive replication is practically infeasible, in particular if inherently small numbers of individuals exhibit any given group effect. Yet it is quite unclear how to conduct inference in this case. We show that inference is possible, even if the number of replications is as small as~2. Two methods are proposed, both based on Fourier inversion. One, which is substantially more computer intensive than the other, exhibits better performance in numerical experiments.