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Multivariate calibration. (English) Zbl 0954.62073
Ghosh, Subir (ed.), Multivariate analysis, design of experiments, and survey sampling. A tribute to Jagdish N. Srivastava. New York, NY: Marcel Dekker. Stat., Textb. Monogr. 159, 265-299 (1999).
In the calibration problem, two variables \(x\) and \(Y\) (possibly vector-valued) are related by \[ Y = g(x) + \varepsilon \] for some function \(g\) and error term \(\varepsilon\). Typically, \(x\) is a more expensive measurement, while \(Y\) is cheaper or easier to obtain. A calibration sample \(\{(x_i,Y_i), i = 1,\ldots,n\}\) of \(n\) independent observations is available, while at the prediction step only \(Y = Y_0\) is observed. The author estimates the corresponding unknown \(x\), denoted by \(\xi\), which satisfies \[ Y_0 = g(\xi) + \varepsilon_0. \] He focuses on the case where \(Y\), and possibly \(x\), are vector-valued. He assumes that the \(x\) observations in the calibration sample have been randomly drawn from the same population as \(\xi\). The problem then essentially becomes one of prediction in multiple regression with some special characteristics. He then deals with point estimation in the alternative case of controlled calibration, in which the experimenter sets the \(x\) values in the calibration sample. Finally, interval estimation in the same context is treated.
For the entire collection see [Zbl 0927.00053].
62H12 Estimation in multivariate analysis
62J07 Ridge regression; shrinkage estimators (Lasso)