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| In [[probability]] and [[statistics]], the '''[[quantile]] function''' specifies, for a given probability in the [[probability distribution]] of a [[random variable]], the value at which the probability of the random variable will be less than or equal to that probability. It is also called the '''percent point function''' or '''inverse cumulative distribution function'''. 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]], the [[cumulative distribution function]] (cdf) and the [[Characteristic function (probability theory)|characteristic function]]. The quantile function, ''Q'', of a probability distribution is the [[inverse function|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.
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| ==Definition==
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| Assuming a continuous and strictly monotonic distribution function,
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| <math>\scriptstyle F\colon R \to (0,1)</math>, the quantile function returns the value below which random draws from the given distribution would fall, (''p''×100) percent of the time. That is, it returns the value of ''x'' such that
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| :<math>F(x) = \Pr(X \le x) = p.\,</math>
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| [[File:Quantile distribution function.svg|thumb|Quantile function for a general distribution function]]
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| If the probability distribution is discrete rather than continuous then there may be gaps between values in the domain of its cdf,
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| while if the cdf is only weakly monotonic there may be "flat spots" in its range. In either case, the quantile function is
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| :<math>Q(p)\,=\,\inf\left\{ x\in R : p \le F(x) \right\} </math>
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| for a probability 0 < ''p'' < 1, and the quantile function returns the minimum value of ''x'' for which the previous probability statement holds.
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| == Simple example ==
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| For example, the quantile function for Exponential(''λ'') (i.e. intensity ''λ'' and [[expected value]] 1/''λ'') is | |
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| :<math>Q(p;\lambda) = \frac{-\ln(1-p)}{\lambda}, \!</math>
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| for 0 ≤ ''p'' < 1. The [[quartile]]s are therefore:
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| ; first quartile : <math>\ln(4/3)/\lambda\,</math>
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| ; [[median]] : <math>\ln(2)/\lambda\,</math>
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| ; third quartile : <math>\ln(4)/\lambda.\,</math>
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| ==Applications==
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| Quantile functions are used in both statistical applications and [[Monte Carlo method]]s.
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| 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]] of an observation whose distribution is known; see the [[quantile]] entry. Before the popularization of computers, it was not uncommon for books to have appendices with statistical tables sampling the quantile function (see, e.g., [http://course.shufe.edu.cn/jpkc/jrjlx/ref/StaTable.pdf]). Statistical applications of quantile functions are discussed extensively by Gilchrist.<ref>{{cite book|author=Gilchrist, W. |year=2000|title=Statistical Modelling with Quantile Functions|isbn=1-58488-174-7}}</ref>
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| Monte-Carlo simulations employ quantile functions to produce non-uniform random or [[pseudorandom number]]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, for example, of simulation methods in modern [[computational finance]] are focusing increasing attention on methods based on quantile functions, as they work well with [[multivariate analysis|multivariate]] techniques based on either [[copula (statistics)|copula]] or quasi-Monte-Carlo methods<ref>{{cite book|author=Jaeckel, P. |year=2002|title=Monte Carlo methods in finance}}</ref> and [[Monte Carlo methods in finance]].
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| ==Calculation==
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| The evaluation of quantile functions often involves [[numerical methods]], as the example of the exponential distribution above is one of the few distributions where a [[closed-form expression]] can be found (others include the [[Uniform distribution (continuous)|uniform]], the [[Weibull distribution|Weibull]], the [[Tukey lambda distribution|Tukey lambda]] (which includes the [[Logistic distribution|logistic]]) and the [[log-logistic distribution|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]] 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]] packages.
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| Quantile functions may also be characterized as solutions of non-linear ordinary and partial [[differential equation]]s. The [[ordinary differential equation]]s for the cases of the [[normal distribution|normal]], [[Student t-distribution|Student]], [[beta distribution|beta]] and [[gamma distribution|gamma]] distributions have been given and solved.<ref>{{cite journal|author=Steinbrecher, G., Shaw, W.T. |year=2008|title=Quantile mechanics|journal=European Journal of Applied Mathematics|volume=19|issue=2|pages=87–112|doi=10.1017/S0956792508007341}}</ref>
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| ==The normal distribution==
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| 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<ref>{{cite journal|author=Wichura, M.J. |year=1988|title=Algorithm AS241: The Percentage Points of the Normal Distribution|journal=Applied Statistics|volume=37|pages=477–484|doi=10.2307/2347330|jstor=2347330|issue=3|publisher=Blackwell Publishing}}</ref> and Acklam (see his web site in External Links). Non-composite rational approximations have been developed by Shaw (see Monte Carlo recycling in External Links).
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| ===Ordinary differential equation for the normal quantile===
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| A non-linear ordinary differential equation for the normal quantile, ''w''(''p''), may be given. It is
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| :<math>\frac{d^2 w}{d p^2} = w \left(\frac{d w}{d p}\right)^2 </math>
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| with the centre (boundary) conditions
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| :<math>w\left(1/2\right) = 0,\, </math>
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| :<math>w'\left(1/2\right) = \sqrt{2\pi}.\, </math>
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| 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).
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| ==The Student's t-distribution==
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| {{main|Student's t-distribution}}
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| 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.<ref>{{cite journal|author=Shaw, W.T. |year=2006|title=Sampling Student’s T distribution – Use of the inverse cumulative distribution function.|journal=Journal of Computational Finance|volume=9|issue=4|pages=37–73}}</ref> The simple cases are as follows:
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| ===ν = 1 (Cauchy distribution)===
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| {{main|Cauchy distribution}}
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| :<math>Q(p) = \tan (\pi(p-1/2)) \!</math>
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| ===ν = 2===
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| :<math>Q(p) = 2(p-1/2)\sqrt{\frac{2}{\alpha}}\!</math>
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| === ν = 4 ===
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| :<math>Q(p) = \operatorname{sign}(p-1/2)\,2\,\sqrt{q-1}\!</math>
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| where
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| :<math>q = \frac{\cos \left( \frac{1}{3} \arccos \left( \sqrt{\alpha} \, \right) \right)}{\sqrt{\alpha}}\!</math>
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| and
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| :<math>\alpha = 4p(1-p).\!</math>
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| 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.
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| ==Quantile mixtures==
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| Analogously to [[mixture distribution|the mixtures of densities]], distributions can be defined as quantile mixtures
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| :<math>Q(p)=\sum_{i=1}^{m}a_i Q_i(p)</math>,
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| where <math>Q_i(p)</math>, <math>i=1,\ldots,m</math> are quantile functions and <math>a_i</math>, <math>i=1,\ldots,m</math> are the model parameters. The parameters <math>a_i</math> must be selected so that <math>Q(p)</math> is a quantile function.
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| Two four-parametric quantile mixtures, the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture, are presented by Karvanen.<ref>{{cite journal|author=Karvanen, J. |year=2006|title=Estimation of quantile mixtures via L-moments and trimmed L-moments.|journal=Computational Statistics & Data Analysis|volume=51|issue=2|pages=947–956|doi=10.1016/j.csda.2005.09.014}}</ref>
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| ==Non-linear differential equations for quantile functions==
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| The non-linear ordinary differential equation given for [[normal distribution]] 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
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| :<math>\frac{d^2 Q}{d p^2} = H(Q) \left(\frac{d Q}{d p}\right)^2 </math> | |
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| augmented by suitable boundary conditions, where
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| :<math> H(x) = -\frac{d \log[f(x)]}{dx} </math> | |
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| 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.
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| == See also ==
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| * [[Inverse transform sampling]]
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| * [[Percent point]]
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| ==References==
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| {{Reflist}}
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| *Abernathy, Roger W. and Smith, Robert P. (1993) *[http://portal.acm.org/citation.cfm?id=168387 "Applying series expansion to the inverse beta distribution to find percentiles of the F-distribution"], ''ACM Trans. Math. Softw.'', 9 (4), 478–480 {{doi|10.1145/168173.168387}}
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| == External links ==
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| *[http://home.online.no/~pjacklam/notes/invnorm/ An algorithm for computing the inverse normal cumulative distribution function]
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| *[http://www.mth.kcl.ac.uk/~shaww/web_page/papers/NormalQuantile1.pdf Refinement of the Normal Quantile]
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| *[http://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf New Methods for Managing "Student's" T Distribution]
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| *[http://portal.acm.org/citation.cfm?id=355600 ACM Algorithm 396: Student's t-Quantiles]
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| *[http://arxiv.org/abs/0901.0638 Computational Finance: Differential Equations for Monte Carlo Recycling]
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| {{Theory of probability distributions}}
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| [[Category:Theory of probability distributions]]
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| [[pt:Quantil]]
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