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Optimal designs for estimating the control values in multi-univariate regression models

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  • Lin, Chun-Sui
  • Huang, Mong-Na Lo

Abstract

This paper considers a linear regression model with a one-dimensional control variable x and an m-dimensional response vector . The components of are correlated with a known covariance matrix. Based on the assumed regression model, it is of interest to obtain a suitable estimation of the corresponding control value for a given target vector on the expected responses. Due to the fact that there is more than one target value to be achieved in the multiresponse case, the m expected responses may meet their target values at different respective control values. Consideration on the performance of an estimator for the control value includes the difference of the expected response E(yi) from its corresponding target value Ti for each component and the optimal value of control point, say x0, is defined to be the one which minimizes the weighted sum of squares of those standardized differences within the range of x. The objective of this study is to find a locally optimal design for estimating x0, which minimizes the mean squared error of the estimator of x0. It is shown that the optimality criterion is equivalent to a c-criterion under certain conditions and explicit solutions with dual response under linear and quadratic polynomial regressions are obtained.

Suggested Citation

  • Lin, Chun-Sui & Huang, Mong-Na Lo, 2010. "Optimal designs for estimating the control values in multi-univariate regression models," Journal of Multivariate Analysis, Elsevier, vol. 101(5), pages 1055-1066, May.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:5:p:1055-1066
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    References listed on IDEAS

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    1. Lo Huang, Mong-Na & Lin, Chun-Sui, 2006. "Minimax and maximin efficient designs for estimating the location-shift parameter of parallel models with dual responses," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 198-210, January.
    2. Krafft, Olaf & Schaefer, Martin, 1992. "D-Optimal designs for a multivariate regression model," Journal of Multivariate Analysis, Elsevier, vol. 42(1), pages 130-140, July.
    3. Rolf Sundberg, 1999. "Multivariate Calibration — Direct and Indirect Regression Methodology," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(2), pages 161-207, June.
    4. Sundberg, Rolf, 1985. "When is the inverse regression estimator MSE-superior to the standard regression estimator in multivariate controlled calibration situations?," Statistics & Probability Letters, Elsevier, vol. 3(2), pages 75-79, April.
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