Incomplete multivariate designs, optimal with respect to Fisher's information matrix
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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.
KeywordsFisher Information Cost Restriction Dispersion Matrix Unpublished Thesis Case R162
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