In
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 ...
, Wilks' lambda distribution (named for
Samuel S. Wilks
Samuel Stanley Wilks (June 17, 1906 – March 7, 1964) was an American mathematician and academic who played an important role in the development of mathematical statistics, especially in regard to practical applications.
Early life and edu ...
), is 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 i ...
used in
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 ...
hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
, especially with regard to the
likelihood-ratio test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after im ...
and
multivariate analysis of variance
In statistics, multivariate analysis of variance (MANOVA) is a procedure for comparing multivariate sample means. As a multivariate procedure, it is used when there are two or more dependent variables, and is often followed by significance tests i ...
(MANOVA).
Definition
Wilks' lambda distribution is defined from two
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
Wishart distributed variables as the
ratio distribution
A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions.
Given two (usually independent) random variables ''X'' ...
of their
determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
s,
[
]
given
:
independent and with
:
where ''p'' is the number of dimensions. In the context of
likelihood-ratio test
In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after im ...
s ''m'' is typically the error degrees of freedom, and ''n'' is the hypothesis degrees of freedom, so that
is the total degrees of freedom.
Approximations
Computations or tables of the Wilks' distribution for higher dimensions are not readily available and one usually resorts to approximations.
One approximation is attributed to
M. S. Bartlett and works for large ''m'' allows Wilks' lambda to be approximated with a
chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
:
Another approximation is attributed to
C. R. Rao
Calyampudi Radhakrishna Rao FRS (born 10 September 1920), commonly known as C. R. Rao, is an Indian-American mathematician and statistician. He is currently professor emeritus at Pennsylvania State University and Research Professor at the Un ...
.
Properties
There is a symmetry among the parameters of the Wilks distribution,
:
Related distributions
The distribution can be related to a product of
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independ ...
beta-distributed random variables
:
:
As such it can be regarded as a multivariate generalization of the beta distribution.
It follows directly that for a one-dimension problem, when the Wishart distributions are one-dimensional with
(i.e., chi-squared-distributed), then the Wilks' distribution equals the beta-distribution with a certain parameter set,
:
From the relations between a beta and an
F-distribution
In probability theory and statistics, the ''F''-distribution or F-ratio, also known as Snedecor's ''F'' distribution or the Fisher–Snedecor distribution (after Ronald Fisher and George W. Snedecor) is a continuous probability distribution th ...
, Wilks' lambda can be related to the F-distribution when one of the parameters of the Wilks lambda distribution is either 1 or 2, e.g.,
:
and
:
See also
*
Chi-squared distribution
In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squa ...
*
Dirichlet distribution
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted \operatorname(\boldsymbol\alpha), is a family of continuous multivariate probability distributions parameterized by a vector \boldsymb ...
*
''F''-distribution
*
Gamma distribution
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distri ...
*
Hotelling's ''T''-squared distribution
*
Student's ''t''-distribution
*
Wishart distribution
In statistics, the Wishart distribution is a generalization to multiple dimensions of the gamma distribution. It is named in honor of John Wishart, who first formulated the distribution in 1928.
It is a family of probability distributions define ...
References
{{statistics, analysis, state=collapsed
Continuous distributions