Pivotal Quantity

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on unknown parameters. Note that a pivot quantity need not be a statistic-the function and its value can depend on parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic. More formally, given an independent and identically distributed sample X = (X_1,X_2,ldots,X_n) from a distribution with parameter ¿, a function g is a pivotal quantity if the distribution of g(X,¿) is independent of ¿. 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.