Mathematical Sciences: Confidence Regions in Multivariate Calibration

DMS 9530932马修 多变量校准的问题可以简单地描述为 如下两种类型的测量可以在每个项目或个人 在一项研究中：其中一个，由向量x表示，是昂贵的，或时间 消耗或有时甚至不可能获得，以及第二测量， 由矢量y表示，更容易获得。校准曲线通过拟合统计模型来构建，所述统计模型建立 y和x之间的关系，基于两个 这些变量。然后将校准曲线用于进一步的研究， 在观察到y之后，关于x的未知值的统计推断。 该研究涉及未知值x的置信区域的构建。需要不同类型的置信区域 这取决于应用。如果校准曲线用于 为x的单个未知值构建置信区域，我们有 一次使用置信区域。另一方面，如果校准 曲线用于构造未知序列的置信区域 x的值，对应于一系列测量y，我们有 多个使用置信区域。同时置信区域可以 当我们有一组给定的未知x值时， 对应于给定的一组独立的y测量。的 将研究这种置信区域的构造 在模型中的各种假设下，y和x。 %校准是化学和工程中广泛使用的程序 以理工科为重校准产生的典型问题涉及样品中化学品浓度的测量或血清样品中药物的量，其中直接测量这些感兴趣的量通常是困难的或昂贵的。间接响应通常很容易获得，并且一旦在响应和感兴趣的量之间建立了关系，当感兴趣的量的值未知时，可以使用该关系艾德来估计感兴趣的量。该研究的目的是获得感兴趣的未知量的区间。特别地，间隔可以用于推断，例如，如果未知浓度 化学品的含量超过了一定的限度。 ***

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. ***