Mathematical Sciences: Confidence Regions in Multivariate Calibration

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DMS 9530932 Mathew The problem of multivariate calibration can be briefly described as follows. Two types of measurements can be made on each item or individual in a study: one of them, denoted by a vector x, is expensive, or time consuming or sometimes even impossible to obtain, and a second measurement, denoted by a vector y, is more easily obtained. A calibration curve is constructed by fitting a statistical model that establishes the relationship between y and x, based on data that is available on both of these variables. The calibration curve is then used in a further study for statistical inference concerning an unknown value of x after observing y. This research involves the construction of confidence regions for the unknown value of x. Different types of confidence regions are needed depending on the application. If the calibration curve is used for constructing a confidence region for a single unknown value of x, we have a single use confidence region. On the other hand, if the calibration curve is used to construct confidence regions for a sequence of unknown values of x, corresponding to a sequence of measurements y, we have multiple use confidence regions. Simultaneous confidence regions may also be required when we have a given set of unknown values of x, corresponding to a given set of independent y measurements. The construction of such confidence regions will be investigated under various assumptions on the model that relates y and x. %%% Calibration is a widely used procedure in the chemical and engineering sciences. Typical problems where calibration arises deal with the measurement of the concentration of a chemical in a sample, or the amount of drug in a serum sample, where the direct measurement of these quantities of interest is usually difficult or expensive. An indirect response is usually easy to obtain and once a relationship is established between the response and the quantity of interest, this relationship can be us ed to estimate the quantity of interest, when its value is unknown. This research is aimed at obtaining an interval for the unknown quantity of interest. In particular, the interval can be used to conclude, for example, if the unknown concentration of the chemical exceeds a certain limit. ***