Quantile function

With reference to a continuous and strictly monotonic cumulative distribution function (c.d.f.)

The quartiles are therefore: Quantile functions are used in both statistical applications and Monte Carlo methods.

Consider a statistical application where a user needs 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.

[2] Statistical applications of quantile functions are discussed extensively by Gilchrist.

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 can be found (others include the uniform, the Weibull, the Tukey lambda (which includes the logistic) and the log-logistic).

[5][6] Further algorithms to evaluate quantile functions are given in the Numerical Recipes series of books.

General methods to numerically compute the quantile functions for general classes of distributions can be found in the following libraries: Quantile functions may also be characterized as solutions of non-linear ordinary and partial differential equations.

The ordinary differential equations for the cases of the normal, Student, beta and gamma distributions have been given and solved.

It is with the centre (initial) conditions This equation may be solved by several methods, including the classical power series approach.

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.

[16] 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.

It is augmented by suitable boundary conditions, where 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.