Quantile function

With reference to a continuous and strictly monotonic cumulative distribution function ${\displaystyle F_{X}\colon \mathbb {R} \to [0,1]}$ of a random variable X, the quantile function ${\displaystyle Q\colon [0,1]\to \mathbb {R} }$ maps its input p to a threshold value x so that the probability of X being less or equal than x is p. In terms of the distribution function F, the quantile function Q returns the value x such that

${\displaystyle F_{X}(x):=\Pr(X\leq x)=p\,,}$

which can be written as inverse of the c.d.f.

${\displaystyle Q(p)=F_{X}^{-1}(p)\,.}$

The cumulative distribution function (shown as F(x)) gives the p values as a function of the q values. The quantile function does the opposite: it gives the q values as a function of the p values. Note that the portion of F(x) in red is a horizontal line segment.

In the general case of distribution functions that are not strictly monotonic and therefore do not permit an inverse c.d.f., the quantile is a (potentially) set valued functional of a distribution function F, given by the interval[1]

${\displaystyle Q(p)\,=\,\left[\sup \left\{x\colon F(x)<p\right\},\sup \left\{x\colon F(x)\leq p\right\}\right]}$

It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of F)

${\displaystyle Q(p)\,=\,\inf \left\{x\in \mathbb {R} :p\leq F(x)\right\}\,.}$

Here we capture the fact that the quantile function returns the minimum value of x from amongst all those values whose c.d.f value exceeds p, which is equivalent to the previous probability statement in the special case that the distribution is continuous. Note that the infimum function can be replaced by the minimum function, since the distribution function is right-continuous and weakly monotonically increasing.

The quantile is the unique function satisfying the Galois inequalities

${\displaystyle Q(p)\leq x}$ if and only if ${\displaystyle p\leq F(x)}$

If the function F is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have:

${\displaystyle Q=F^{-1}}$

In general, even though the distribution function F may fail to possess a left or right inverse, the quantile function Q behaves as an "almost sure left inverse" for the distribution function, in the sense that

${\displaystyle Q(F(X))=X}$ almost surely.

The normal distribution is perhaps the most important case. Because the normal distribution is a location-scale family, its quantile function for arbitrary parameters can be derived from a simple transformation of the quantile function of the standard normal distribution, known as the probit function. Unfortunately, this function has no closed-form representation using basic algebraic functions; as a result, approximate representations are usually used. Thorough composite rational and polynomial approximations have been given by Wichura[6] and Acklam.[7] Non-composite rational approximations have been developed by Shaw.[8]

A non-linear ordinary differential equation for the normal quantile, w(p), may be given. It is

${\displaystyle {\frac {d^{2}w}{dp^{2}}}=w\left({\frac {dw}{dp}}\right)^{2}}$

with the centre (initial) conditions

${\displaystyle w\left(1/2\right)=0,\,}$

${\displaystyle w'\left(1/2\right)={\sqrt {2\pi }}.\,}$

This equation may be solved by several methods, including the classical power series approach. From this solutions of arbitrarily high accuracy may be developed (see Steinbrecher and Shaw, 2008).

This has historically been one of the more intractable cases, as the presence of a parameter, ν, the degrees of freedom, makes the use of rational and other approximations awkward. Simple formulas exist when the ν = 1, 2, 4 and the problem may be reduced to the solution of a polynomial when ν is even. In other cases the quantile functions may be developed as power series.[9] The simple cases are as follows:

ν = 1 (Cauchy distribution)

${\displaystyle Q(p)=\tan(\pi (p-1/2))\!}$

ν = 2

${\displaystyle Q(p)=2(p-1/2){\sqrt {\frac {2}{\alpha }}}\!}$

ν = 4

${\displaystyle Q(p)=\operatorname {sign} (p-1/2)\,2\,{\sqrt {q-1}}\!}$

where

${\displaystyle q={\frac {\cos \left({\frac {1}{3}}\arccos \left({\sqrt {\alpha }}\,\right)\right)}{\sqrt {\alpha }}}\!}$

and

${\displaystyle \alpha =4p(1-p).\!}$

In the above the "sign" function is +1 for positive arguments, −1 for negative arguments and zero at zero. It should not be confused with the trigonometric sine function.