In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.
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| - Pivotstatistik (de)
- Pivotal quantity (en)
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| - Eine Pivotstatistik, auch Pivot-Größe genannt, kurz ein Pivot, ist eine spezielle Funktion in der mathematischen Statistik. Es handelt sich um Statistiken mit bestimmten Invarianzeigenschaften, die zur Konstruktion von Konfidenzbereichen verwendet werden. Der Name leitet sich ab vom französischen pivot (deutsch Anker, hier im Sinne von Dreh- und Angelpunkt) und beruht auf den Invarianzeigenschaften. (de)
- In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic. (en)
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| - Eine Pivotstatistik, auch Pivot-Größe genannt, kurz ein Pivot, ist eine spezielle Funktion in der mathematischen Statistik. Es handelt sich um Statistiken mit bestimmten Invarianzeigenschaften, die zur Konstruktion von Konfidenzbereichen verwendet werden. Der Name leitet sich ab vom französischen pivot (deutsch Anker, hier im Sinne von Dreh- und Angelpunkt) und beruht auf den Invarianzeigenschaften. (de)
- In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). A pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic. More formally, let be a random sample from a distribution that depends on a parameter (or vector of parameters) . Let be a random variable whose distribution is the same for all . Then is called a pivotal quantity (or simply a pivot). Pivotal quantities are commonly used for normalization to allow data from different data sets to be compared. It is relatively easy to construct pivots for location and scale parameters: for the former we form differences so that location cancels, for the latter ratios so that scale cancels. Pivotal quantities are fundamental to the construction of test statistics, as they allow the statistic to not depend on parameters – for example, Student's t-statistic is for a normal distribution with unknown variance (and mean). They also provide one method of constructing confidence intervals, and the use of pivotal quantities improves performance of the bootstrap. In the form of ancillary statistics, they can be used to construct frequentist prediction intervals (predictive confidence intervals). (en)
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