Wilks's 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 and multivariate analysis of variance (MANOVA). Wilks' lambda distribution is defined from two independent Wishart distributed variables as the ratio distribution of their determinants, given independent and with where p is the number of dimensions. 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 is the total degrees of freedom. 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 Another approximation is attributed to C. R. Rao. There is a symmetry among the parameters of the Wilks distribution, The distribution can be related to a product of independent 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, 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.

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Related concepts (7)

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. Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles"), or Wishart–Laguerre ensemble (since its eigenvalue distribution involve Laguerre polynomials), or LOE, LUE, LSE (in analogy with GOE, GUE, GSE).

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 that arises frequently as the null distribution of a test statistic, most notably in the analysis of variance (ANOVA) and other F-tests. The F-distribution with d1 and d2 degrees of freedom is the distribution of where and are independent random variables with chi-square distributions with respective degrees of freedom and .

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