In

"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 Quantile

New Methods for Managing "Student's" T Distribution

ACM Algorithm 396: Student's t-Quantiles

{{Theory of probability distributions Functions related to probability distributions pt:Quantil

probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the quantile
In statistics and probability, quantiles are cut points dividing the Range (statistics), range of a probability distribution into continuous intervals with equal probabilities, or dividing the Observation (statistics), observations in a Sample (s ...

function, associated with a probability distribution
In probability theory and statistics, a probability distribution is the mathematical Function (mathematics), function that gives the probabilities of occurrence of different possible outcomes for an Experiment (probability theory), experiment. ...

of a random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

, 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 monotoniccumulative 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 ...

$F\_X\backslash colon\; \backslash mathbb\; \backslash to;\; href="/html/ALL/s/,1.html"\; ;"title=",1">,1$random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...

''X'', the quantile function $Q\backslash colon;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$ 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
:$F\_X(x)\; :=\; \backslash Pr(X\; \backslash le\; x)\; =\; p\backslash ,,$
which can be written as inverse of the c.d.f.
:$Q(p)\; =F\_X^(p)\backslash ,.$
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 :$Q(p)\backslash ,=\backslash ,\backslash left;\; href="/html/ALL/s/sup\backslash left\backslash ,\_\backslash sup\backslash left\backslash \backslash right.html"\; ;"title="sup\backslash left\backslash ,\; \backslash sup\backslash left\backslash \backslash right">sup\backslash left\backslash ,\; \backslash sup\backslash left\backslash \backslash right$ It is often standard to choose the lowest value, which can equivalently be written as (using right-continuity of ''F'') :$Q(p)\backslash ,=\backslash ,\backslash inf\backslash left\backslash \; \backslash ,.$ 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 :$Q(p)\; \backslash le\; x$ if and only if $p\; \backslash le\; F(x)$ 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'' andexpected 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 l ...

(mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude (mathematics), magnitude and sign (mathematics), sign) of a gi ...

) 1/''λ'') is
:$F(x;\backslash lambda)\; =\; \backslash begin\; 1-e^\; \&\; x\; \backslash ge\; 0,\; \backslash \backslash \; 0\; \&\; x\; <\; 0.\; \backslash end$
The quantile function for Exponential(''λ'') is derived by finding the value of Q for which $1-e^\; =p$:
:$Q(p;\backslash lambda)\; =\; \backslash frac,\; \backslash !$
for 0 ≤ ''p'' < 1. The quartile
In statistics, a quartile is a type of quantile which divides 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 a ...

s are therefore:
; first quartile (p = 1/4): $-\backslash ln(3/4)/\backslash lambda\backslash ,$
; median
In statistics and probability theory, the median 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 th ...

(p = 2/4) : $-\backslash ln(1/2)/\backslash lambda\backslash ,$
; third quartile (p = 3/4) : $-\backslash ln(1/4)/\backslash lambda.\backslash ,$
Applications

Quantile functions are used in both statistical applications andMonte 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 determini ...

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), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ...

(pdf) or probability mass function
In probability theory, 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. T ...

, 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 (mathematics), function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'' ...

. 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.
For statistical applications, users need 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, but a ...

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 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 (statistics), range of a probability distribution into continuous intervals with equal probabilities, or dividing the Observation (statistics), observations in a Sample (s ...

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 Statistical randomness, statistically random, despite having been produced by a completely Deterministic system, deterministic and repeatable process.
Background
The generation of ran ...

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, 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) financial instrument, instruments, portfolio (finance), portfolios and investments by simulation, simulating the various sources of uncertainty ...

.
Calculation

The evaluation of quantile functions often involvesnumerical methods
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

, 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 expression (mathematics), mathematical expression that uses a finite set, finite number of standard operations. It may contain Constant (mathematics), constants, Variable (mathematics), variables, c ...

can be found (others include the uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

, 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 mathematics and computing, a root-finding algorithm is an algorithm for finding Zero of a function, zeros, also called "roots", of continuous functions. A zero of a function , from the real numbers to real numbers or from the complex numbers to ...

such as the bisection method
In mathematics, the bisection method is a Root-finding algorithm, root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly Bisection, bisecting the Interva ...

to invert the cdf. Other algorithms to evaluate quantile functions are given in the Numerical Recipes series of books. Algorithms for common distributions are built into many statistical software
Statistical software are specialized computer program
A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. Computer programs are one component of software, which als ...

packages.
Quantile functions may also be characterized as solutions of non-linear ordinary and partial differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more unknown function (mathematics), functions and their derivatives. In applications, the functions generally represent physical quantities, the der ...

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 (mathematics), variable and involves the derivatives of those functions. The term ''ordinary ...

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 The Commonwealth, commonwealth countries, a "student" attends a secondary school or higher (e.g., college or university); those in pri ...

, beta
Beta (, ; uppercase , lowercase , or cursive Greek, 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 th ...

and gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician a ...

distributions have been given and solved.
Normal distribution

Thenormal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The param ...

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 Q–Q plot, exploratory statistical gra ...

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 :$\backslash frac\; =\; w\; \backslash left(\backslash frac\backslash right)^2$ with the centre (initial) conditions :$w\backslash left(1/2\backslash right)\; =\; 0,\backslash ,$ :$w\text{'}\backslash left(1/2\backslash right)\; =\; \backslash sqrt.\backslash ,$ 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) :$Q(p)\; =\; \backslash tan\; (\backslash pi(p-1/2))\; \backslash !$ ;ν = 2 :$Q(p)\; =\; 2(p-1/2)\backslash sqrt\backslash !$ ;ν = 4 :$Q(p)\; =\; \backslash operatorname(p-1/2)\backslash ,2\backslash ,\backslash sqrt\backslash !$ where :$q\; =\; \backslash frac\backslash !$ and :$\backslash alpha\; =\; 4p(1-p).\backslash !$ 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 :$Q(p)=\backslash sum\_^a\_i\; Q\_i(p)$, where $Q\_i(p)$, $i=1,\backslash ldots,m$ are quantile functions and $a\_i$, $i=1,\backslash ldots,m$ 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 fornormal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The param ...

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
:$\backslash frac\; =\; H(Q)\; \backslash left(\backslash frac\backslash right)^2$
augmented by suitable boundary conditions, where
:$H(x)\; =\; -\backslash frac\; =\; -\backslash frac\; \backslash ln\; f(x)$
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
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, but a ...

* Probability integral transform
* Quantile
In statistics and probability, quantiles are cut points dividing the Range (statistics), range of a probability distribution into continuous intervals with equal probabilities, or dividing the Observation (statistics), observations in a Sample (s ...

* Rank–size distribution
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 Quantile

New Methods for Managing "Student's" T Distribution

ACM Algorithm 396: Student's t-Quantiles

{{Theory of probability distributions Functions related to probability distributions pt:Quantil