Sankhya: The Indian Journal of Statistics

1994, Volume 56, Series B, Pt. 1, pp. 121--126

ON THE ASYMPTOTIC PRECISION OF THE WELCH APPROACH

By

R. ARDANUY and J. F. LOPEZ, Universidad de Salamanca

SUMMARY. In this paper we attempt to investigate the order of accuracy in Welch type approximations of the distributions of certain studentized statistics. We compare the exact cumulative distribution function of statistic $T = \frac{X-{\theta}}{\sqrt{\sum{\lambda}_{i} S_2^{i}}}$ with the value obtained by means of Welch approximation. The main result is that this difference is Op(f-3/2) and also Op(f-2) in some sampling cases, where $f = \frac{{(\sum{\lambda_{i}\sigma_i^2})}^{2}}{\sum{\frac{\lambda_{i}^{2}\sigma_{i}^{4}}{f_i}}}$ and f1,, fk are the degrees of freedom of unbiased estimators of variances. In particular we find that in several important hypotheses testing problems, the asymptotic order of the error in Welch approximation is greatly reduced if we take the sample sizes proportional to the respective population standard deviations.

AMS subject classification. 62F05, 62F03, 62E20.

Key words and phrases. Asymptotic properties of tests, Behrens-Fisher problem, difference of means, hypothesis testing, Welch approximation.

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