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| In [[statistics]], a '''pivotal quantity''' or '''pivot''' is a function of observations and unobservable parameters whose [[probability distribution]] does not depend on the unknown [[parameter]]s <ref>Shao, J.: ''Mathematical Statistics'', Springer, New York, 2003, ISBN 978-0-387-95382-3 (Section 7.1)</ref> (also referred to as [[nuisance parameter]]s). 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,<ref>Morris H. DeGroot, Mark J. Schervish: ''Probability and Statistics'' (4th Edition), Pearson, 2011 (page 489)</ref> let <math>X = (X_1,X_2,\ldots,X_n) </math> be a random sample from a distribution that depends on a parameter (or vector of parameters) <math> \theta </math>. Let <math> g(X,\theta) </math> be a random variable whose distribution is the same for all <math> \theta </math>. Then <math>g</math> is called a ''pivotal quantity'' (or simply a ''pivotal'').
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| Pivotal quantities are commonly used for [[Normalization (statistics)|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.
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| Pivotal quantities are fundamental to the construction of [[test statistic]]s, 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 interval]]s, and the use of pivotal quantities improves performance of the [[bootstrapping (statistics)|bootstrap]]. In the form of ancillary statistics, they can be used to construct frequentist [[prediction interval]]s (predictive confidence intervals).
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| == Examples ==
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| === Normal distribution ===
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| {{see also|Prediction interval#Normal distribution}}
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| One of the simplest pivotal quantities is the [[z-score]]; given a normal distribution with <math>\mu</math> and variance <math>\sigma^2</math>, and an observation ''x,'' the z-score:
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| : <math> z = \frac{x - \mu}{\sigma},</math>
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| has distribution <math>N(0,1)</math> – a normal distribution with mean 0 and variance 1. Similarly, since the ''n''-sample sample mean has sampling distribution <math>N(\mu,\sigma^2/n),</math> the z-score of the mean
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| : <math> z = \frac{\overline{X} - \mu}{\sigma/\sqrt{n}}</math>
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| also has distribution <math>N(0,1).</math> Note that while these functions depend on the parameters – and thus one can only compute them if the parameters are known (they are not statistics) – the distribution is independent of the parameters.
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| Given <math>n</math> independent, identically distributed (i.i.d.) observations <math>X = (X_1, X_2, \ldots, X_n) </math> from the [[normal distribution]] with unknown mean <math>\mu</math> and variance <math>\sigma^2</math>, a pivotal quantity can be obtained from the function:
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| :<math> g(x,X) = \sqrt{n}\frac{x - \overline{X}}{s} </math>
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| where
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| :<math> \overline{X} = \frac{1}{n}\sum_{i=1}^n{X_i} </math>
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| and
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| :<math> s^2 = \frac{1}{n-1}\sum_{i=1}^n{(X_i - \overline{X})^2} </math>
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| are unbiased estimates of <math>\mu</math> and <math>\sigma^2</math>, respectively. The function <math>g(x,X)</math> is the [[Student's t-statistic]] for a new value <math>x</math>, to be drawn from the same population as the already observed set of values <math>X</math>.
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| Using <math>x=\mu</math> the function <math>g(\mu,X)</math> becomes a pivotal quantity, which is also distributed by the [[Student's t-distribution]] with <math>\nu = n-1</math> degrees of freedom. As required, even though <math>\mu</math> appears as an argument to the function <math>g</math>, the distribution of <math>g(\mu,X)</math> does not depend on the parameters <math>\mu</math> or <math>\sigma</math> of the normal probability distribution that governs the observations <math>X_1,\ldots,X_n</math>.
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| This can be used to compute a [[prediction interval]] for the next observation <math>X_{n+1};</math> see [[Prediction interval#Normal distribution|Prediction interval: Normal distribution]].
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| === Bivariate normal distribution ===
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| In more complicated cases, it is impossible to construct exact pivots. However, having approximate pivots improves convergence to [[asymptotic normality]].
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| Suppose a sample of size <math>n</math> of vectors <math>(X_i,Y_i)'</math> is taken from a bivariate [[normal distribution]] with unknown [[correlation]] <math>\rho</math>.
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| An estimator of <math>\rho</math> is the sample (Pearson, moment) correlation
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| :<math> r = \frac{\frac1{n-1} \sum_{i=1}^n (X_i - \overline{X})(Y_i - \overline{Y})}{s_X s_Y} </math>
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| where <math>s_X^2, s_Y^2</math> are [[sample variance]]s of <math>X</math> and <math>Y</math>. The sample statistic <math>r</math> has an asymptotically normal distribution:
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| :<math>\sqrt{n}\frac{r-\rho}{1-\rho^2} \Rightarrow N(0,1)</math>.
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| However, a [[variance-stabilizing transformation]]
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| :<math> z = \rm{tanh}^{-1} r = \frac12 \ln \frac{1+r}{1-r}</math>
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| known as [[Fisher transformation|Fisher's ''z'' transformation]] of the correlation coefficient allows to make the distribution of <math>z</math> asymptotically independent of unknown parameters:
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| :<math>\sqrt{n}(z-\zeta) \Rightarrow N(0,1)</math>
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| where <math>\zeta = {\rm tanh}^{-1} \rho</math> is the corresponding population parameter. For finite samples sizes <math>n</math>, the random variable <math>z</math> will have distribution closer to normal than that of <math>r</math>. An even closer approximation to the standard normal distribution is obtained by using a better approximation for the exact variance: the usual form is
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| :<math>\operatorname{Var}(z) \approx \frac1{n-3} .</math>
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| == Robustness ==
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| {{main|Robust statistics}}
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| 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.
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| 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.
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| == See also ==
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| * [[Normalization (statistics)]]
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| == References ==
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| {{Reflist}}
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| [[Category:Statistical theory]]
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