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

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## Summary

Consider a*p*-variate normal population*N* _{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 matrix*H*, corresponding to the estimation of the variances*σ* _{tt}. Consider the marginal involving the responses*i* _{1},*i* _{2}, …,*i* _{k}, and suppose that from this marginal a sample of*n*(*i* _{1},*i* _{2}, …,*i* _{k}) is drawn. It is then seen that*H* is a linear function of the*n*'s. Suppose that the cost of taking an observation on the*j*th response is*φ* _{0}, and that a total amount of money*φ* _{′} is available for the collection of samples. The problem considered in this paper is the following. How to choose the*n*'s subject to the cost restriction, such that the determinant of*H* is maximized. A complete solution is obtained for the case*p*=2. When*p*=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Fisher Information Cost Restriction Dispersion Matrix Unpublished Thesis Case R162## Preview

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### References

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