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In probability and statistics, 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 th ...
function, associated with a
probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...
of a random variable, specifies the value of the random variable such that the probability of the variable being less than or equal to that value equals the given probability. Intuitively, the quantile function associates with a range at and below a probability input the likelihood that a random variable is realized in that range for some probability distribution. It is also called the percentile function, percent-point function or inverse cumulative distribution function.


Definition


Strictly monotonic distribution function

With reference to a continuous and strictly monotonic
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 ...
F_X\colon \mathbb \to ,1/math> of a random variable ''X'', the quantile function Q\colon
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to \mathbb returns a threshold value ''x'' below which random draws from the given c.d.f. would fall ''100*p'' percent of the time. In terms of the distribution function ''F'', the quantile function ''Q'' returns the value ''x'' 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 ''F'', given by the interval :Q(p)\,=\,\left sup\left\, \sup\left\\right It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of ''F'') :Q(p)\,=\,\inf\left\ \,. 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 :Q(p) \le x if and only if p \le F(x) If the function ''F'' 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 ''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 : Q(F(X))=X almost surely.


Simple example

For example, the cumulative distribution function of Exponential(''λ'') (i.e. intensity ''λ'' and
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
( mean) 1/''λ'') is :F(x;\lambda) = \begin 1-e^ & x \ge 0, \\ 0 & x < 0. \end The quantile function for Exponential(''λ'') is derived by finding the value of Q for which 1-e^ =p : :Q(p;\lambda) = \frac, \! for 0 ≤ ''p'' < 1. The quartiles are therefore: ; first quartile (p = 1/4): -\ln(3/4)/\lambda\, ; median (p = 2/4) : -\ln(1/2)/\lambda\, ; third quartile (p = 3/4) : -\ln(1/4)/\lambda.\,


Applications

Quantile functions are used in both statistical applications and Monte Carlo methods. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function (pdf) or
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
, 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 point ...
. The quantile function, ''Q'', of a probability distribution is the
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...
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. For statistical applications, users need to know key percentage points 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 it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
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 th ...
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 numbers 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, are focusing increasing attention on methods based on quantile functions, as they work well with
multivariate Multivariate may refer to: In mathematics * Multivariable calculus * Multivariate function * Multivariate polynomial In computing * Multivariate cryptography * Multivariate division algorithm * Multivariate interpolation * Multivariate optical c ...
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 dis ...
.


Calculation

The evaluation of quantile functions often involves numerical methods, such as the exponential distribution above, which is one of the few distributions where a
closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
can be found (others include the uniform, 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 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 th ...
to invert the cdf. Other algorithms to evaluate quantile functions are given in the
Numerical Recipes ''Numerical Recipes'' is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1 ...
series of books. Algorithms for common distributions are built into many
statistical software Statistical software are specialized computer programs for analysis in statistics and econometrics. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software ...
packages. Quantile functions may also be characterized as solutions of non-linear ordinary and partial
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. The
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s for the cases of the normal,
Student A student is a person enrolled in a school or other educational institution. In the United Kingdom and most commonwealth countries, a "student" attends a secondary school or higher (e.g., college or university); those in primary or elementar ...
,
beta Beta (, ; uppercase , lowercase , or cursive ; grc, βῆτα, bē̂ta or ell, βήτα, víta) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Modern Greek, it represents the voiced labiod ...
and gamma distributions have been given and solved.


Normal distribution

The
normal distribution In 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^ The parameter \mu i ...
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 s ...
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, ''w''(''p''), 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 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 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. 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_^a_i Q_i(p), where Q_i(p), i=1,\ldots,m are quantile functions and a_i, 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 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^ The parameter \mu i ...
is a special case of that available for any quantile function whose second derivative exists. In general the equation for a quantile, ''Q''(''p''), 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 ''ƒ''(''x'') 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, Smirnov transform, or the golden ruleAalto University, N. Hyvönen, Computational methods in inverse probl ...
* Percentage point * Probability integral transform *
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 th ...
*
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 a ...


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