Formalized by
John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its
quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in
statistical model
A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, ...
s directly.
The Tukey lambda distribution has a single
shape parameter, λ, and as with other probability distributions, it can be transformed with a
location parameter, μ, and a
scale parameter, σ. Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.
Quantile function
For the standard form of the Tukey lambda distribution, the quantile function,
(i.e. the inverse function to 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 ...
) and the quantile density function (
are
:
:
For most values of the shape parameter, , the
probability density function (PDF) and
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) must be computed numerically. The Tukey lambda distribution has a simple, closed form for the CDF and / or PDF only for a few exceptional values of the shape parameter, for example: (see
uniform distribution
Uniform distribution may refer to:
* Continuous uniform distribution
* Discrete uniform distribution
* Uniform distribution (ecology)
* Equidistributed sequence
See also
*
* Homogeneous distribution
In mathematics, a homogeneous distribution ...
ase = 1and the
logistic distribution
Logistic may refer to:
Mathematics
* Logistic function, a sigmoid function used in many fields
** Logistic map, a recurrence relation that sometimes exhibits chaos
** Logistic regression, a statistical model using the logistic function
** Logit, ...
ase = 0.
However, for any value of both the CDF and PDF can be tabulated for any number of cumulative probabilities, , using the quantile function to calculate the value , for each cumulative probability , with the probability density given by , the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table.
Moments
The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for and is given by the formula (except when ''λ'' = 0)
:
More generally, the ''n''-th order moment is finite when and is expressed in terms of the
beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral
: \Beta(z_1,z_2) = \int_0^1 t^( ...
''Β''(''x'',''y'') (except when ''λ'' = 0) :
:
Note that due to symmetry of the density function, all moments of odd orders are equal to zero.
L-moments
Differently from the central moments,
L-moments
In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics ( L-statistics) analogous to conventional moments, and can be used to calculate qu ...
can be expressed in a closed form. The L-moment of order ''r>1'' is given by
:
The first six L-moments can be presented as follows:
:
:
:
:
:
:
Comments
The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
The most common use of this distribution is to generate a Tukey lambda
PPCC plot of a
data set A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more database tables, where every column of a table represents a particular variable, and each row corresponds to a given record of th ...
. Based on the
PPCC plot, an appropriate
model
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
for the data is suggested. For example, if the best-fit of the curve to the data occurs for a value of at or near 0.14, then the data could be well-modeled with a normal distribution. Values of less than 0.14 suggests a heavier-tailed distribution; a milepost at = 0 (logistic) would indicate quite fat tails, with the extreme limit at = −1, approximating Cauchy. That is, as the best-fit value of varies from 0.14 towards −1, a bell-shaped PDF with increasingly heavy tails is suggested. Similarly, for an optimal value of becomes greater than 0.14 suggests a distribution with ''exceptionally'' thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with).
Except for values of very close to 0, all the suggested PDF functions have finite
support, between and .
Since the Tukey lambda distribution is a
symmetric
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A
histogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.
References
External links
*
{{ProbDistributions, continuous-variable
Continuous distributions
Probability distributions with non-finite variance