Estimation of two normal means which may be common. (English) Zbl 0716.62032
Summary: Consider the problem of estimating the mean of a normal population when independent samples from this as well as a second normal population are available. Pre-test estimators, which combine the two sample means if a test of the hypothesis of equal population means accepts but otherwise use only the first sample mean, are compared to limited translation estimators which are derived in the spirit of P. J. Bickel [Ann. Stat. 12, 864-879 (1984; Zbl 0545.62028)] (we also cover the cases of unknown variances).
Our conclusion is that if the accuracy with which the second population mean can be estimated is of the same or better order of magnitude as the accuracy with which the first can be estimated, then the limited translation estimators largely dominate the pre-test estimators in terms of mean square error loss.
Our conclusion is that if the accuracy with which the second population mean can be estimated is of the same or better order of magnitude as the accuracy with which the first can be estimated, then the limited translation estimators largely dominate the pre-test estimators in terms of mean square error loss.
MSC:
| 62F10 | Point estimation |
Keywords:
common mean; Graybill-Deal estimator; problem P; normal population; Pre- test estimators; test of the hypothesis of equal population means; limited translation estimators; unknown variances; mean square error lossCitations:
Zbl 0545.62028References:
| [1] | Bickel, P. J. (1984). Parametric robustness: small biases can be worthwhile, Ann. Statist., 12, 864-879. · Zbl 0545.62028 · doi:10.1214/aos/1176346707 |
| [2] | Efron, B. and Morris, C. (1971). Limiting the risk of Bayes and empirical Bayes estimators?Part I: the Bayes case, J. Amer. Statist. Assoc., 66, 807-815. · Zbl 0229.62003 · doi:10.2307/2284231 |
| [3] | Graybill, F. A. and Deal, R. B. (1959). Combining unbiased estimators, Biometrics, 15, 543-550. · Zbl 0096.34503 · doi:10.2307/2527652 |
| [4] | Ohtani, K. (1987). On the use of the Graybill-Deal estimator when two normal population means may not be common, Comm. Statist. B?Simulation Comput., 16, 855-870. · doi:10.1080/03610918708812622 |
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