SEQUENTIAL FIXED-PRECISION ESTIMATION IN STOCHASTIC LINEAR REGRESSION MODELS

ABSTRACT

The goal of this article is to address the problem of fixed-precision estimation of parameters in a linear regression setup with stochastic regressors. First in the case of a linear regression model where the regressor variable and the random errors have independent Gaussian distributions, sequential sampling schemes are proposed for fixed proportional accuracy estimation and fixed-width interval estimation of the regression-slope based on a Chebyshev inequality approach. The asymptotic second-order efficiency of these procedures is then established using the techniques developed in Aras and Woodroofe (1993)[2]. Asymptotic second-order expansions are derived for a lower bound of P (relative error < preassigned bound) in the fixed proportional accuracy estimation case and that of the coverage-probability in the interval-estimation case. In both cases, the possibility of relaxing the Gaussian assumption is explored, leading to a reconsideration of Martinsek's (1995)[15] fixed proportional accuracy estimation and a detailed discussion of a stochastic multiple linear regression model with distribution-free errors and regressors. In this distribution-free multiple regression scenario, construction of fixed-size confidence regions for the vector of regression-parameters is considered and an asymptotically second-order efficient sequential methodology is put forward. Moderate sample-size performances of some of these procedures are investigated via simulation-studies.

Keywords:

• Stochastic linear regression
• Sequential sampling
• Second-order asymptotics
• Fixed-proportional-accuracy estimation
• Chebyshev's inequality
• AMS 1991 Subject Classification
• 62F10
• 62F12
• 62F25
• 62G05

ACKNOWLEDGMENTS

This research was supported by National Science Foundation grants NSF-G-ASC-9504041 and NSF-DMS-9157715. The author is grateful to Prof. Michael Woodroofe for his useful suggestions and an anonymous referee for a long list of helpful remarks. Thanks also to prof. Nitis Mukhopadhyay and Dr. Janis Hardwick for their comments on the clarity of presentation and overall organization of the paper.