Wilks's lambda distribution

In statistics, Wilks's 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).

Definition

Wilks's lambda distribution is defined from two independent Wishart distributed variables as the ratio distribution of their determinants,^[1]

given

\mathbf{A} \sim W_p(\Sigma, m) \qquad \mathbf{B} \sim W_p(\Sigma, n)

independent and with $m \ge p$

\lambda = \frac{\det(\mathbf{A})}{\det(\mathbf{A+B})} = \frac{1}{\det(\mathbf{I}+\mathbf{A}^{-1}\mathbf{B})} \sim \Lambda(p,m,n)

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 $n+m$ is the total degrees of freedom.^[1]

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 m^[2] allows Wilks's lambda to be approximated with a chi-squared distribution

\left(\frac{p-n+1}{2}-m\right)\log \Lambda(p,m,n) \sim \chi^2_{np}.

^[1]

Another approximation is attributed to C R Rao.^[1]^[3]

Properties

There is a symmetry among the parameters of the Wilks distribution,^[1]

\Lambda(p, m, n) \sim \Lambda(n, m + n - p, p)

Related distributions

The distribution can be related to a product of independent beta-distributed random variables

u_i \sim B\left(\frac{m+1-i}{2},\frac{n}{2}\right)

\prod_{i=1}^p u_i \sim \Lambda(p,m,n).

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 $p=1$ (i.e., chi-squared-distributed), then the Wilks' distribution equals the beta-distribution with a certain parameter set,

\Lambda(1,m,n) \sim B\left(\frac{m}{2},\frac{n}{2}\right).

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.,^[1]

\frac{1 - \Lambda(p, m, 1)}{\Lambda(p, m, 1)} \sim \frac{p}{m - p + 1} F_{p, m-p+1},

and

\frac{1 - \sqrt{\Lambda(p, m, 2)}}{\sqrt{\Lambda(p, m, 2)}} \sim \frac{p}{m - p + 1} F_{2p, 2(m-p+1)}.

References

1 2 3 4 5 6 Kanti Mardia, John T. Kent and John Bibby (1979). Multivariate Analysis. Academic Press. ISBN 0-12-471250-9.
↑ M. S. Bartlett (1954). "A Note on the Multiplying Factors for Various $\chi^2$ Approximations". J R Stat Soc Series B 16 (2): 296–298. JSTOR 2984057. delete character in |title= at position 47 (help)
↑ C R Rao (1951). "An asymptotic expansion of the distribution of Wilks' criterion". Bulletin de l'Institut International de Statistique 33: 177–180.

Probability distributions

List of probability distributions

Discrete univariate with finite support	Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine Reciprocal Triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory-Schulz Fréchet Gamma Gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type Poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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