The Career of a Research Statistician pp 33-46 | Cite as

# Estimating the Common Parameter of Several Distributions

- 35 Downloads

## Abstract

As described in Sect. 1.3, the consulting problem was that of estimating the common mean of two normal distributions, when the data of the two samples are given, and the corresponding variances are unknown. There are two types of problems. One is an estimation problem, when the samples from the two distributions are given. In this case we are in the estimation phase. Prior to this phase there is a design phase, in which one has to determine how many observations to draw from each distribution. Obviously, if we know which distribution has the smaller variance, then the best is to draw all the observations from that distribution. If this information is not available, the design problem becomes very difficult if the objective is to minimize the total number of observations, under the constraint that the variance (or mean-squared error) of the estimator satisfies certain condition. This design problem is similar to the well-known “two-armed bandit problem.” The present chapter is devoted to the estimation problem, when two or more samples of equal size are given from the corresponding distributions. For a sequential design problem, when the value of one variance is known, see the paper of Zacks (Naval Res Logist Q 21:569–574, 1974).

## References

- Brown, L. D., & Cohen, A. (1974). Point and confidence estimation of a common mean and recovery of inter-block information.
*The Annals of Statistics, 2*, 963–976.MathSciNetCrossRefGoogle Scholar - Cohen, A. (1976). Combining estimates of location.
*Journal of the American Statistical Association, 71*, 172–175.MathSciNetCrossRefGoogle Scholar - Cohen, A., & Sackrowitz, H. B. (1974). On estimating the common mean of two normal distributions.
*The Annals of Statistics, 2*, 1274–1282.MathSciNetCrossRefGoogle Scholar - George, E. (1991). Shrinkage domination in a multivariate common mean problem.
*The Annals of Statistics, 19*, 952–960.MathSciNetCrossRefGoogle Scholar - Ghezzi, D., & Zacks, S. (2005). Inference on the common variance of correlated normal random variables.
*Communication in Statistics-Theory and Methods, 34*, 1517–1531.MathSciNetCrossRefGoogle Scholar - Graybill, F. A., & Deal, R. B. (1959). Combining unbiased estimators.
*Biometrics, 15*, 543–550.MathSciNetCrossRefGoogle Scholar - Graybill, F. A., & Weeks, D. L. (1959). Combining inter-block and intra-block in incomplete blocks.
*Annals of Mathematical Statistics, 30*, 799–805.MathSciNetCrossRefGoogle Scholar - Hanner, D. M., & Zacks, S. (2013). On two-stage sampling for fixed-width interval estimation of the common variance of equi-correlated normal distributions.
*Sequential Analysis, 32*, 1–13.MathSciNetCrossRefGoogle Scholar - Keller, T., & Olkin, I. (2004). Combining correlated unbiased estimators of the mean of a Normal distribution.
*A Festschrift for Herman Rubin, IMS Lecture Notes-Monographs, Series, 45*, 218–227.MathSciNetCrossRefGoogle Scholar - Kubokawa, T. (1987a). Estimation of the common mean of normal distributions with applications to regression and design of experiments. Ph.D. Dissertation, University of Tsukuba.Google Scholar
- Kubokawa, T. (1987b). Admissible minimax estimators of a common mean of two normal populations.
*Annals of Statistics, 15*, 1245–1256.MathSciNetCrossRefGoogle Scholar - Zacks, S. (1966a). Randomized fractional weighing designs.
*The Annals of Mathematical Statistics, 37*, 1382–1395.CrossRefGoogle Scholar - Zacks, S. (1966b). Unbiased estimation of the common mean of two normal distributions based on small samples.
*Journal of the American Statistical Association, 61*, 467–476.MathSciNetCrossRefGoogle Scholar - Zacks, S. (1970a). Bayesian design of single and double stratified sampling for estimating proportion finite populations.
*Technometrics, 12*, 119–130.CrossRefGoogle Scholar - Zacks, S. (1970b). Bayes and fiducial equivariant estimators of the common mean to two normal distributions.
*The Annals of Mathematical Statistics, 41*, 59–67.MathSciNetCrossRefGoogle Scholar - Zacks, S. (1974). On the optimality of the Bayes prediction policy in two-echelon multi-station inventory systems.
*Naval Research Logistics Quarterly, 21*, 569–574.MathSciNetCrossRefGoogle Scholar - Zacks, S. (2014).
*Problems and examples in mathematical statistics*. New York: Wiley.zbMATHGoogle Scholar - Zacks, S., & Ramig, P. A. (1987). Confidence intervals for the common variance of equi-correlated normal random variables. In A. E. Gelfand (Ed.)
*Contributions for the theory and applications of statistics*(pp. 511–544). A Volume in Honor of Herbert Solomon. New York: Academic.Google Scholar