Very weak expansions for sequentially designed experiments.
Ruby C. Weng · Deep Blue (University of Michigan) · 1999
Sequential methods were used to solve testing problems more efficiently. But at the same time, they make statistical inference more difficult, because they affect the sampling distributions of estimators. Typically, the effect on the sampling distributions is largest for small and moderate expected sample sizes and decreases as the expected sample sizes increases; but the rate of decrease may be slow. In this dissertation, we study the problem of setting confidence intervals following a sequential test. One main result provides approximate confidence intervals that are valid to a high order of magnitude for sequential samples from a multiparameter exponential family. This result is illustrated by an application to group sequential testing of two Poisson means. Simulations for this special case are conducted and they compare well with the theoretical results. The construction of these confidence sets starts with a data dependent transformation of the parameters that converts the likelihood function into a normal form. This transformation provides a first approximation to a pivotal quantity. To a high order of magnitude, the first approximation is asymptotically normal with a mean and covariance matrix that are slightly different from 0 and the identity matrix. This leads to a renormalized variable that is then shown to be asymptotically standard normal to a high order of magnitude. Constructing confidence sets from the latter is a simple process. The mathematical derivation also has application to time series. A second main result is the construction of corrected confidence sets for the parameters of stationary autoregressive processes. The quality of the approximations is checked by comparing the theoretical results to the simulation results in an autoregressive process of order two. The corrected pivotal quantity significantly performs better. The methods have still other potential applications. Some of these are described.