*J. Japan Statist. Soc.,* Vol. 42 (No. 2), pp. 165-184, 2012

## ASYMPTOTIC EXPANSION OF THE PERCENTILES FOR A SAMPLE MEAN STANDARDIZED BY GMD IN A NORMAL CASE WITH APPLICATIONS

Nitis Mukhopadhyay and Bhargab Chattopadhyay

**Abstract. **
This paper develops an asymptotic expansion of a percentile point of the Gini- based standardized sample mean. Such approximate percentiles can be used for proposing tests of hypotheses or confidence intervals of μ when samples arrive from a normal distribution with unknown mean μ and standard deviation σ. We have asymptotically expressed the percentile point *b*_{m,α} of the Gini-based pivot (1.5), that is, the Gini-based standardized sample mean. Using large-scale simulations, approximations, and data analyses, we report that the Gini-based test and confidence interval procedures for μ perform better or practically as well as the customarily employed Student’s *t*-based procedures when samples arrive from a normal distribution with suspect outliers. This interesting finding is especially noteworthy when we have a small random sample from a normal population with possible outliers.

*Key words and phrases*:
Cornish-Fisher expansion, Gini's mean difference, normalized sample mean, outlier, robustness, suspect outlier, taylor expansion.

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