In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on the unknown parameters (also referred to as nuisance parameters). Note that 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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More formally, let
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).
Normal distribution
One of the simplest pivotal quantities is the z-score; given a normal distribution with mean
has distribution
also has distribution
Given
where
and
are unbiased estimates of
Using
This can be used to compute a prediction interval for the next observation
Bivariate normal distribution
In more complicated cases, it is impossible to construct exact pivots. However, having approximate pivots improves convergence to asymptotic normality.
Suppose a sample of size
An estimator of
where
However, a variance-stabilizing transformation
known as Fisher's z transformation of the correlation coefficient allows to make the distribution of
where
Robustness
From the point of view of robust statistics, pivotal quantities are robust to changes in the parameters – indeed, independent of the parameters – but not in general robust to changes in the model, such as violations of the assumption of normality. This is fundamental to the robust critique of non-robust statistics, often derived from pivotal quantities: such statistics may be robust within the family, but are not robust outside it.