Sequential Confidence Regions in Inverse Regression Problems

Abstract

In inverse regression problems (or more generally, the estimation of ratios of regression parameters) and errors-in-variables models, it has been shown by Gleser and Hwang that the length of any confidence interval with positive confidence level is infinite with positive probability. Therefore the confidence sets derived using asymptotic theory, although having correct asymptotic coverage probability, typically have zero confidence level when the sample size is fixed. Is it possible to construct a sequential confidence interval with finite length and $1 - \alpha > 0$ confidence level? The answer is no for any finite stage sequential sampling. The answer is, however, yes for a fully sequential scheme, as demonstrated by Hwang and Liu. For the inverse regression problem, and more generally the set estimation of a ratio of regression parameters, we construct a $(1 - \alpha)$ confidence sequence. Applying such a confidence sequence, we can construct a $(1 - \alpha)$ sequential confidence interval with the length less than a prespecified quantity.