Wilks' lambda distribution
Encyclopedia
In statistics
, Wilks' lambda distribution (named for Samuel S. Wilks
), is a probability distribution
used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test
and Multivariate analysis of variance. It is a multivariate generalization of the univariate F-distribution, and generalizes Hotelling's T-squared distribution in the same way that the F-distribution generalizes Student's t-distribution.
Wilks' lambda distribution is related to two independent
Wishart distributed variables, and is
defined as follows,
given
independent and with
where p is the number of dimensions. In the context of likelihood-ratio test
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.
The distribution can be related to a product of independent
Beta distributed random variables
For large m Bartlett's approximation
allows Wilks' lambda to be approximated with a Chi-squared distribution
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, Wilks' lambda distribution (named for Samuel S. Wilks
Samuel S. Wilks
Samuel Stanley Wilks was an American mathematician and academic who played an important role in the development of mathematical statistics, especially in regard to practical applications....
), is a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
used in multivariate hypothesis testing, especially with regard to the likelihood-ratio test
Likelihood-ratio test
In statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which is a special case of the other . The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other...
and Multivariate analysis of variance. It is a multivariate generalization of the univariate F-distribution, and generalizes Hotelling's T-squared distribution in the same way that the F-distribution generalizes Student's t-distribution.
Wilks' lambda distribution is related to two independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
Wishart distributed variables, and is
defined as follows,
given
independent and with
where p is the number of dimensions. In the context of likelihood-ratio test
Likelihood-ratio test
In statistics, a likelihood ratio test is a statistical test used to compare the fit of two models, one of which is a special case of the other . The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other...
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.The distribution can be related to a product of independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
Beta distributed random variables
For large m Bartlett's approximation
allows Wilks' lambda to be approximated with a Chi-squared distribution