A sequential approach to fixed-width, asymptotic confidence intervals for ratios of parameters in linear regression.

Abstract

Sequential methods are applied to the Inverse Regression (Calibration) Problem. Fixed width, asymptotic confidence intervals are set for ratios of parameters in two regression models. The first procedure sets intervals for 1/$\beta$, from the linear regression model with slope $\beta$ and an intercept set to zero. The second procedure sets intervals for $\alpha/\beta$, from the linear regression model with slope $\beta$ and an intercept $\alpha$. The experimenter sets both the desired coverage probability and the length of the interval. As the lengths of the intervals are set closer to zero, the actual coverage probability converges to the desired coverage probability. These procedures are unique in that the coverage probability converges uniformly for $\beta \in (0,\beta\sp*),\beta\sp* > 0.$ Uniformity in $\beta$ down to zero is not attainable in fixed sample size procedures. In fact, any confidence interval for 1/$\beta$ or $\alpha/\beta$ where $\beta \in (0,\beta\sp*),$ based on a fixed sample size, will have either infinite expected length or zero confidence, confidence being the infimum (over the parameter set) of the coverage probability, (Gleser and Hwang, 1987, Ann. Statistics). The predictors may be deterministic, random or adaptive. Adaptive designs allow the predictors to be functions of the previously observed data. In particular, the predictors may be functions of the best estimate of $\beta$. The errors, in the regression model, are assumed to be independent and identitically distributed random variables with mean zero, positive variance and have finite moments of order greater then two. The estimators of 1/$\beta$ and $\alpha$/$\beta$ have finite moments, the order determined by the number of moments assumed on the errors. The first problem, that of setting an interval for 1/$\beta$, is simplified by embedding the regression through the origin model into a Brownian motion with drift $\beta$. The regression data is then viewed as observations taken at discrete time points of a Brownian motion with drift $\beta$. Hence the problem is reduced to setting intervals for the reciprocal of the drift, which is considered by Keener and Woodroofe, (1992, J. Statist. Planning and Inference). The embedding is accomplished by applying Strassen's strong approximation result for martingales, (Strassen, 1965, Proc. Fifth Berkeley Symposium Math. Statist. Prob.). The techniques developed in the first problem are applied to the second problem and are expected to be applicable to ratios of parameters in other linear regression models and to the errors-in-variables model.