Annals of the Institute of Statistical Mathematics

, Volume 26, Issue 1, pp 299–313

# Incomplete multivariate designs, optimal with respect to Fisher's information matrix

• J. N. Srivastava
• M. K. Zaatar
Article

## Summary

Consider ap-variate normal populationN p(μ, Σ), with μ known and Σ unknown. Without loss of generality, we take μ=0. Suppose that we have an incomplete multiresponse sample, i.e., we have samples available from this population and/or its various marginals. Suppose one is interested in estimating Σ, given that all the correlations are known.

Consider the Fisher information matrixH, corresponding to the estimation of the variancesσ tt. Consider the marginal involving the responsesi 1,i 2, …,i k, and suppose that from this marginal a sample ofn(i 1,i 2, …,i k) is drawn. It is then seen thatH is a linear function of then's. Suppose that the cost of taking an observation on thejth response isφ 0, and that a total amount of moneyφ is available for the collection of samples. The problem considered in this paper is the following. How to choose then's subject to the cost restriction, such that the determinant ofH is maximized. A complete solution is obtained for the casep=2. Whenp=3, some partial results are obtained, in particular, it is shown that when all the costs are equal, and the correlations are equal, then the best design is obtained by using a complete sample.

## Keywords

Fisher Information Cost Restriction Dispersion Matrix Unpublished Thesis Case R162
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Hocking, R. R. and Smith, W. B. (1971). Optimal-incomplete multinormal samples, To appear inTechnometrics.Google Scholar
2. [2]
Kleinbaum, D. G. (1970). Estimation and hypothesis testing for generalized multivariate linear models, Unpublished thesis, University of North Carolina, Chapel Hill, N.C.Google Scholar
3. [3]
Marcus, M. and Minc, H. (1964).A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston.
4. [4]
Rao, C. R. (1965).Linear Statistical Inference and Its Applications, Wiley and Sons, New York.
5. [5]
Roy, S. N., Gnanadesikan, R. and Srivastava, J. N. (1971).Analysis and Design of Certain Quantitative Multiresponse Experiments, Pergamon Press, New York.
6. [6]
Srivastava, J. N. and McDonald, L. L. (1969). On the costwise optimality of hierarchical multiresponse randomized block designs under the trace criterion,Ann. Inst. Statist. Math.,21, 507–514.
7. [7]
Srivastava, J. N. and McDonald, L. L. (1971). On the cost wise optimality of certain hierarchical and standard multiresponse models under the determinant criterion,J. Multivariate Analysis, 1.Google Scholar
8. [8]
Srivastava, J. N. and McDonald, L. L. (1970). On the hierarchical two-response (cyclic PBIB) designs, costwise optimal under the trace criterion,Ann. Inst. Statist. Math.,22, 507–518.
9. [9]
Trawinsky, I. M. (1961). Incomplete variable designs, Unpublished thesis, Virginia Polytechnic Institute, Blacksburg, Virginia.Google Scholar