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probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the quantile function is a function Q: ,1\mapsto \mathbb which maps some probability x \in ,1/math> of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
v to the value of the variable y such that P(v\leq y) = x according to its
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
. In other words, the function returns the value of the variable below which the specified cumulative probability is contained. For example, if the distribution is a
standard normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac e^ ...
then Q(0.5) will return 0 as 0.5 of the probability mass is contained below 0. The quantile function is also called the percentile function (after the
percentile In statistics, a ''k''-th percentile, also known as percentile score or centile, is a score (e.g., a data point) a given percentage ''k'' of all scores in its frequency distribution exists ("exclusive" definition) or a score a given percentage ...
), percent-point function, inverse cumulative distribution function (after the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
or c.d.f.) or inverse distribution function.


Definition


Strictly increasing distribution function

With reference to a continuous and strictly increasing
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
(c.d.f.) F_X\colon \mathbb \to ,1/math> of a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
, the quantile function Q\colon
, 1 The comma is a punctuation mark that appears in several variants in different languages. Some typefaces render it as a small line, slightly curved or straight, but inclined from the vertical; others give it the appearance of a miniature fille ...
\to \mathbb maps its input ''p'' to a threshold value so that the probability of being less or equal than is . In terms of the distribution function , the quantile function returns the value such that F_X(x) := \Pr(X \le x) = p, which can be written as inverse of the c.d.f. Q(p) = F_X^(p).


General distribution function

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 , given by the interval Q(p) = \big sup\, \sup\\big It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of ) Q(p) = \inf \. Here we capture the fact that the quantile function returns the minimum value of from amongst all those values whose c.d.f value exceeds , which is equivalent to the previous probability statement in the special case that the distribution is continuous. The quantile is the unique function satisfying the Galois inequalities Q(p) \le x if and only if p \le F(x). If the function is continuous and strictly monotonically increasing, then the inequalities can be replaced by equalities, and we have Q = F^. In general, even though the distribution function may fail to possess a left or right inverse, the quantile function behaves as an "almost sure left inverse" for the distribution function, in the sense that Q\bigl(F(X)\bigr) = X \quad \text


Simple example

For example, the cumulative distribution function of Exponential(''λ'') (i.e. intensity and
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
(
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
) ) is F(x; \lambda) = \begin 1 - e^ & x \ge 0, \\ 0 & x < 0. \end The quantile function for is derived by finding the value of for which 1 - e^ = p: Q(p; \lambda) = \frac, for . The
quartile In statistics, quartiles are a type of quantiles which divide the number of data points into four parts, or ''quarters'', of more-or-less equal size. The data must be ordered from smallest to largest to compute quartiles; as such, quartiles are ...
s are therefore: ; first quartile (): -\ln(3/4) / \lambda, ;
median The median of a set of numbers is the value separating the higher half from the lower half of a Sample (statistics), data sample, a statistical population, population, or a probability distribution. For a data set, it may be thought of as the “ ...
() : -\ln(1/2) / \lambda, ; third quartile () : -\ln(1/4) / \lambda.


Applications

Quantile functions are used in both statistical applications and
Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be ...
s. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
(pdf) or
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
, the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
(cdf) and the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts: * The indicator function of a subset, that is the function \mathbf_A\colon X \to \, which for a given subset ''A'' of ''X'', has value 1 at points ...
. The quantile function, ''Q'', of a probability distribution is the inverse of its cumulative distribution function ''F''. The derivative of the quantile function, namely the quantile density function, is yet another way of prescribing a probability distribution. It is the reciprocal of the pdf composed with the quantile function. Consider a statistical application where a user needs to know key
percentage point A percentage point or percent point is the unit (measurement), unit for the difference (mathematics), arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points (altho ...
s of a given distribution. For example, they require the median and 25% and 75% quartiles as in the example above or 5%, 95%, 2.5%, 97.5% levels for other applications such as assessing the
statistical significance In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
of an observation whose distribution is known; see the
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
entry. Before the popularization of computers, it was not uncommon for books to have appendices with statistical tables sampling the quantile function. Statistical applications of quantile functions are discussed extensively by Gilchrist. Monte-Carlo simulations employ quantile functions to produce non-uniform random or
pseudorandom number A pseudorandom sequence of numbers is one that appears to be statistically random, despite having been produced by a completely deterministic and repeatable process. Pseudorandom number generators are often used in computer programming, as tradi ...
s for use in diverse types of simulation calculations. A sample from a given distribution may be obtained in principle by applying its quantile function to a sample from a uniform distribution. The demands of simulation methods, for example in modern
computational finance Computational finance is a branch of applied computer science that deals with problems of practical interest in finance.Rüdiger U. Seydel, ''Tools for Computational Finance'', Springer; 3rd edition (May 11, 2006) 978-3540279235 Some slightly diff ...
, are focusing increasing attention on methods based on quantile functions, as they work well with multivariate techniques based on either copula or quasi-Monte-Carlo methods and
Monte Carlo methods in finance Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze (complex) instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the di ...
.


Calculation

The evaluation of quantile functions often involves
numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
, such as the exponential distribution above, which is one of the few distributions where a
closed-form expression In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
can be found (others include the
uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
, the Weibull, the Tukey lambda (which includes the logistic) and the log-logistic). When the cdf itself has a closed-form expression, one can always use a numerical
root-finding algorithm In numerical analysis, a root-finding algorithm is an algorithm for finding zeros, also called "roots", of continuous functions. A zero of a function is a number such that . As, generally, the zeros of a function cannot be computed exactly nor ...
such as the
bisection method In mathematics, the bisection method is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval defined by these values and t ...
to invert the cdf. Other methods rely on an approximation of the inverse via interpolation techniques. Further algorithms to evaluate quantile functions are given in the
Numerical Recipes ''Numerical Recipes'' is the generic title of a series of books on algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a cla ...
series of books. Algorithms for common distributions are built into many
statistical software The following is a list of statistical software. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software suite for non-linear statistical modeling based on C+ ...
packages. General methods to numerically compute the quantile functions for general classes of distributions can be found in the following libraries: * C library UNU.RAN * R library Runuran * Python subpackage sampling in scipy.stats Quantile functions may also be characterized as solutions of non-linear ordinary and partial differential equations. The
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s for the cases of the normal,
Student A student is a person enrolled in a school or other educational institution, or more generally, a person who takes a special interest in a subject. In the United Kingdom and most The Commonwealth, commonwealth countries, a "student" attends ...
,
beta Beta (, ; uppercase , lowercase , or cursive ; or ) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represe ...
and
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
distributions have been given and solved.


Normal distribution

The
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
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 In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and ...
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 and Acklam. Non-composite rational approximations have been developed by Shaw.


Ordinary differential equation for the normal quantile

A non-linear ordinary differential equation for the normal quantile, , may be given. It is \frac = w \left(\frac\right)^2 with the centre (initial) conditions w\left(1/2\right) = 0,\, w'\left(1/2\right) = \sqrt.\, This equation may be solved by several methods, including the classical
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
approach. From this solutions of arbitrarily high accuracy may be developed (see Steinbrecher and Shaw, 2008).


Student's ''t''-distribution

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 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. The simple cases are as follows: ;ν = 1 (Cauchy distribution) : Q(p) = \tan (\pi(p-1/2)) \! ;ν = 2 : Q(p) = 2(p-1/2)\sqrt\! ;ν = 4 : Q(p) = \operatorname(p-1/2)\,2\,\sqrt\! where q = \frac\! and \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.


Quantile mixtures

Analogously to the mixtures of densities, distributions can be defined as quantile mixtures Q(p) = \sum_^m a_i Q_i(p), where Q_i(p), i = 1,\ldots,m are quantile functions and i=1,\ldots,m are the model parameters. The parameters a_i must be selected so that Q(p) is a quantile function. Two four-parametric quantile mixtures, the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture, are presented by Karvanen.


Non-linear differential equations for quantile functions

The non-linear ordinary differential equation given for
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
is a special case of that available for any quantile function whose second derivative exists. In general the equation for a quantile, , may be given. It is \frac = H(Q) \left(\frac\right)^2 augmented by suitable boundary conditions, where H(x) = -\frac = -\frac \ln f(x) and is the probability density function. The forms of this equation, and its classical analysis by series and asymptotic solutions, for the cases of the normal, Student, gamma and beta distributions has been elucidated by Steinbrecher and Shaw (2008). Such solutions provide accurate benchmarks, and in the case of the Student, suitable series for live Monte Carlo use.


See also

*
Inverse transform sampling Inverse transform sampling (also known as inversion sampling, the inverse probability integral transform, the inverse transformation method, or the Smirnov transform) is a basic method for pseudo-random number sampling, i.e., for generating sampl ...
*
Percentage point A percentage point or percent point is the unit (measurement), unit for the difference (mathematics), arithmetic difference between two percentages. For example, moving up from 40 percent to 44 percent is an increase of 4 percentage points (altho ...
*
Probability integral transform In probability theory, the probability integral transform (also known as universality of the uniform) relates to the result that data values that are modeled as being random variables from any given continuous distribution can be converted to rando ...
*
Quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
*
Rank–size distribution Rank–size distribution is the distribution of size by rank, in decreasing order of size. For example, if a data set consists of items of sizes 5, 100, 5, and 8, the rank-size distribution is 100, 8, 5, 5 (ranks 1 through 4). This is also known ...


References


Further reading

*Abernathy, Roger W. and Smith, Robert P. (1993)
"Applying series expansion to the inverse beta distribution to find percentiles of the F-distribution"
''ACM Trans. Math. Softw.'', 9 (4), 478–480
Refinement of the Normal QuantileNew Methods for Managing "Student's" T DistributionACM Algorithm 396: Student's t-Quantiles
{{Theory of probability distributions Functions related to probability distributions pt:Quantil