Wilks' lambda distribution

Wilks' lambda distribution

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. It is a generalization of the F-distribution, and generalizes Hotelling's T-square 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 isdefined as follows,cite book
last = Mardia
first = K.V.
coauthors = J.T. Kent, J.M. Bibby
title = Multivariate Analysis
publisher = Academic Press
date = 1979
]

given

:A sim W_p(I, m) qquad B sim W_p(I, n)

independent and with m ge p

:lambda = frac sim Lambda(p,m,n).

The distribution can be related to a product of independent Beta distributed random variables:u_i sim Bleft(frac{m+i-p}{2},frac{p}{2} ight):prod_{i=1}^n u_i sim Lambda(p,m,n).

In the context of likelihood-ratio tests "m" is typically the error degrees of freedom, and "n" is the hypothesis degrees of freedom, so that n+m is the total degrees of freedom.

For large "m" Bartlett's approximation [cite journal
last = Bartlett
first = M.S.
title = A note on multiplying factors for various chi^2 approximations
journal = J. Royal Statist. Soc. Series B
volume = 16
pages = 296–298
date = 1954
] allows Wilks' lambda to be approximated with a Chi-square distribution:left(frac{p-n+1}{2}-m ight)log Lambda(p,m,n) sim chi^2_{np}.

References


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Wilks-Lambda test —   Statistical test comparing the centroids of a distribution of means; a multivariate analysis of variance.   See also ANOVA …   Expanded glossary of Cycad terms

  • Wilks — may refer to:* Alan Wilks * Eileen Wilks * Ella Wilks * Guy Wilks * Henry Wilks * Ivor Wilks * Jim Wilks * John Wilks * Marian Wilks * Mark Wilks * Mark Wilks Collet * Maurice Wilks * Mike Wilks (author) * Mike Wilks (basketball) * Nicholas Wilks …   Wikipedia

  • Samuel S. Wilks — Samuel Stanley Wilks (June 17, 1906 ndash; 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.Born in Little Elm,… …   Wikipedia

  • Hotelling's T-square distribution — In statistics, Hotelling s T square statistic, [ H. Hotelling (1931) The generalization of Student s ratio , Ann. Math. Statist., Vol. 2, pp 360 ndash;378.] named for Harold Hotelling,is a generalization of Student s t statistic that is used in… …   Wikipedia

  • Ratio distribution — A ratio distribution (or quotient distribution ) is a statistical distribution constructed as the distribution of the ratio of random variables having two other distributions.Given two stochastic variables X and Y , the distribution of the… …   Wikipedia

  • Exponential distribution — Not to be confused with the exponential families of probability distributions. Exponential Probability density function Cumulative distribution function para …   Wikipedia

  • Noncentral chi-squared distribution — Noncentral chi squared Probability density function Cumulative distribution function parameters …   Wikipedia

  • Negative binomial distribution — Probability mass function The orange line represents the mean, which is equal to 10 in each of these plots; the green line shows the standard deviation. notation: parameters: r > 0 number of failures until the experiment is stopped (integer,… …   Wikipedia

  • Normal-gamma distribution — Normal gamma parameters: location (real) (real) (real) (real) support …   Wikipedia

  • Conway–Maxwell–Poisson distribution — Conway–Maxwell–Poisson parameters: support: pmf: cdf …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”