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).
[1] A pivot need not be a statistic — the function and its value can depend on the parameters of the model, but its distribution must not.
be a random variable whose distribution is the same for all
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).
One of the simplest pivotal quantities is the z-score.
, and an observation 'x', the z-score: has distribution
– a normal distribution with mean 0 and variance 1.
independent, identically distributed (i.i.d.)
, a pivotal quantity can be obtained from the function: where and are unbiased estimates of
is the Student's t-statistic for a new value
, to be drawn from the same population as the already observed set of values
becomes a pivotal quantity, which is also distributed by the Student's t-distribution with
appears as an argument to the function
of the normal probability distribution that governs the observations
This can be used to compute a prediction interval for the next observation
see Prediction interval: 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
is taken from a bivariate normal distribution with unknown correlation
is the sample (Pearson, moment) correlation where
has an asymptotically normal distribution: However, a variance-stabilizing transformation known as Fisher's 'z' transformation of the correlation coefficient allows creating the distribution of
asymptotically independent of unknown parameters: where
For finite samples sizes
An even closer approximation to the standard normal distribution is obtained by using a better approximation for the exact variance: the usual form is 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.