Generalized inverse distribution function

The (generalized) inverse distribution function , also called quantile transformation or quantile function , is a special real function in stochastics , a branch of mathematics . Each distribution function can be assigned a generalized inverse distribution function which, under certain conditions, is the inverse function of the distribution function. The generalized inverse distribution function assigns every number between zero and one the smallest value at which the distribution function exceeds that number.

If, for example, a probability distribution describes the shoe sizes of Europeans and the corresponding distribution function is given, the associated generalized inverse distribution function specifies the shoe size at that point, so that more than 90% of Europeans wear a shoe size smaller than . ${\displaystyle 0{,}9}$ ${\displaystyle s}$ ${\displaystyle s}$

The generalized inverse distribution function is used, among other things, to determine quantiles . It also provides an approach for the construction of random variables with given distributions . Following the same underlying idea, it is used in the inversion method to generate random numbers with a predetermined distribution from standard random numbers .

definition

Be

${\displaystyle F\colon \mathbb {R} \to \mathbb {R} }$

a distribution function in the sense that it grows monotonically and is continuous on the right-hand side as well as has the limit value behavior and . ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ ${\displaystyle \lim _{x\to \infty }F(x)=1}$

Then the function is called

${\displaystyle F^{-1}\colon (0,1)\to \mathbb {R} }$

defined by

${\displaystyle F^{-1}(u):=\inf\{x\in \mathbb {R} \mid F(x)\geq u\}}$

the generalized inverse distribution function of . ${\displaystyle F}$

Comments on the definition

It should be noted that the distribution function for which the generalized inverse distribution function is defined does not necessarily have to belong to a probability distribution . It only has to fulfill the four properties mentioned above (monotony, legal continuity and the two limit value properties). This is because the generalized inverse distribution function is used to construct probability distributions with a distribution function . To require the existence of such a probability distribution in the definition would be circular. ${\displaystyle F}$

The notation of the generalized inverse distribution function as is to be understood as suggestive, since the distribution function does not always have to be invertible . This occurs, for example, when it is constant over an interval. However, if it is invertible, the inverse of the distribution function and the generalized inverse distribution function agree. Since the generalized inverse distribution function always exists in contrast to the inverse, this justifies the designation as "generalized". ${\displaystyle F^{-1}}$ ${\displaystyle F}$ ${\displaystyle F}$

Explanation

By definition, the function value of the generalized inverse distribution function is the smallest number at the point where the distribution function exceeds the function value . ${\displaystyle u}$ ${\displaystyle u}$

If the distribution function is continuous, this value can be clearly obtained in the following way: A straight line parallel to the x-axis is drawn, which is shifted upwards by the value . This intersects the distribution function at a point or an interval. If it intersects the distribution function at a point , the function value of the generalized inverse distribution function is at that point . If the straight line intersects the distribution function in an interval, the point from the interval that has the smallest coordinate is selected. ${\displaystyle u}$ ${\displaystyle (x,u)}$ ${\displaystyle x}$ ${\displaystyle u}$ ${\displaystyle x}$

example

As an example, consider the distribution function of the exponential distribution . It is given by

${\displaystyle F(x)={\begin{cases}1-\mathrm {e} ^{-\lambda x}&x\geq 0,\\0&x<0,\end{cases}}}$

where is a truly positive real parameter. It grows to strictly monotonously and maps this interval bijectively to . Thus there is an unambiguous inverse function , which can be achieved by resolving ${\displaystyle \lambda }$ ${\displaystyle (0,\infty )}$ ${\displaystyle (0,1)}$ ${\displaystyle F^{-1}}$

${\displaystyle u=1-\mathrm {e} ^{-\lambda x}}$

according to results. This gives the generalized inverse distribution function ${\displaystyle x}$

${\displaystyle F^{-1}(u)={\frac {-\ln(1-u)}{\lambda }}}$.

In general, it is seldom possible to directly compute the generalized inverse distribution function as here. Very few distribution functions can be inverted because they often have constant ranges. An example of this are the distribution functions of discrete distributions . Likewise, even in the case of invertibility, there does not have to be a closed representation of the distribution function that could be used. The distribution function of the normal distribution must always be calculated numerically.

properties

The generalized inverse distribution function is monotonically growing, continuous on the left and thus a random variable or measurable from to . If the measuring space is provided with the constant uniform distribution or, equivalent to the Lebesgue measure , then the following applies: ${\displaystyle ((0,1),{\mathcal {B}}((0,1)))}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle ((0,1),{\mathcal {B}}((0,1)))}$ ${\displaystyle {\mathcal {U}}_{(0,1)}}$

The distribution of under ${\displaystyle F^{-1}}$ ${\displaystyle {\mathcal {U}}_{(0,1)}}$ is the probability measure on which the distribution function has. ${\displaystyle \mathbb {R} }$ ${\displaystyle F}$

Each probability measure on with distribution function may thus as distribution of the random variable ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle F_{P}}$

${\displaystyle F_{P}^{-1}\colon ((0,1),{\mathcal {B}}((0,1)),{\mathcal {U}}_{(0,1)})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

be understood.

use

Construction of random variables of a given distribution

Random variables are introduced as measurable maps between measurement rooms. If a probability measure is also defined on the base space, its distribution can be defined. In the course of further abstraction, however, the basic space and the associated probability measure become less and less important in contrast to the distribution of the random variable. It can effectively be shown that for every random variable with a given distribution a suitable basic space can be supplemented with a probability measure. The generalized inverse distribution function provides such an argument for real distributions: Every real-valued random variable with a given distribution can be understood as a random variable on the interval from zero to one, provided with the constant uniform distribution. Thus, the investigation of random variables and their distributions can be detached from the underlying probability space.

Construction of stochastically independent random variables

The above construction is partly used to show the existence of real-valued independent random variables . First, the existence of stochastically independent random variables independent of the interval is shown using an approximation argument. The concatenation of these random variables with predetermined generalized inverse distribution functions are then real-valued random variables with a predetermined distribution and again stochastically independent. ${\displaystyle (0,1)}$

Determination of quantiles

Is a probability distribution (or a random variable with distribution ) is given, provides the corresponding generalized inverse distribution function , evaluated at the point , always a - quantile . This follows directly from the definition. ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle P_{X}=P}$ ${\displaystyle F^{-1}}$ ${\displaystyle u}$ ${\displaystyle u}$

literature

• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .

Individual evidence

1. ↑ ^{a b} Kusolitsch: Measure and probability theory. 2014, p. 113.
2. ^ Georgii: Stochastics. 2009, p. 23.
3. ↑ Eric W. Weisstein : Quantile Function . In: MathWorld (English).
4. ↑ Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 43 , doi : 10.1007 / 978-3-642-36018-3 .
5. ^ Georgii: Stochastics. 2009, pp. 72-73.