The Wishart distribution is a probability distribution. It is the multivariate equivalent of the χ2 - distribution.
For an explanation for better understanding is first assumed to be random variables: one considers a standard normal random variable X, ie with the expected value 0 and variance 1 There are of these variables n observations or realizations xi (i = 1, ..., n ) ago. Since the realizations take place independently of each other, we interpret them as a sequence of n standard normal random variables Xi. The square sum of these random variables
Is then χ2 -distributed with n degrees of freedom. Summarizing the observations xi in a vector x with n elements together, one can also represent y as the norm
Where xT is a row vector.
It p many different random variables Xj are now considered. These random variables are jointly normally distributed with mean 0 and covariance matrix Σ. There are for each random variable before each n many observations. We can now summarize these data in a ( n × p ) matrix X:
As above, it is the symmetric matrix with the elements
This matrix W is now Wishart distribution with n degrees of freedom.
Properties of the Wishart distribution
As the χ2 - distribution is the Wishart distribution reproductively: The sum of p Wishart - distributed random variable with n degrees of freedom and p random variable with m degrees of freedom is again a total of Wishart distribution with m n degrees of freedom.
Discrete univariate distributions for finite sets: Benford | Bernoulli | beta - binomial | binomial | categorical | hypergeometric | Rademacher | generalized binomial | Zipf | Zipf - Mandelbrot
Discrete univariate distributions for infinite sets: Boltzmann | Conway - Maxwell - Poisson | negative binomial | extended negative binomial | Compound Poisson | discrete uniform | discrete phase -type | Gauss - Kuzmin | geometric | logarithmic | parabolic fractal | Poisson | Poisson - Gamma | Skellam | Yule- Simon | Zeta
Continuous univariate distributions with compact interval: Beta | Cantor | Kumaraswamy | raised cosine | Triangle | U - square | steady uniform | Wigner semicircle
Continuous univariate distributions with half-open interval: Beta prime | Bose -Einstein | Burr | Chi-Square | Coxian | Erlang | Exponential | F | Fermi -Dirac | Folded normal | Fréchet | Gamma | Gamma Gamma | extreme | generalized inverse Gaussian | semi logistically | semi- normal | Hotelling's T-square | hyper- exponential | hypoexponential | inverse chi-square | scale - inverse- chi-square | inverse Normal | inverse gamma | Levy | log-normal | log- logistically | Maxwell -Boltzmann | Maxwell speed | Nakagami | not centered chi-square | Pareto | Phase -Type | Rayleigh | relativistic Breit-Wigner | Rice | Rosin -Rammler | shifted Gompertz | truncated normal | Type -2 Gumbel | Weibull | Wilks ' lambda
Continuous univariate distributions with unbounded interval: Cauchy | extreme | exponentially Power | Fishers z | Fisher - Tippett ( Gumbel ) | generalized hyperbolic | Hyperbolic- secant | Landau | Laplace | alpha- stable | logistics | normal ( Gaussian ) | normal - inverse Gauß'sch | skew - normal | Student's t | Type -1 Gumbel | Variance gamma | Voigt
Discrete multivariate distributions: Ewen | multinomial | Dirichlet compound multinomial
Continuous multivariate distributions: Dirichlet | generalized Dirichlet | multivariate normal | multivariate Student | normal scaled inverse gamma | Normal - Gamma
Multivariate matrix distributions: Inverse Wishart | matrix normal | Wishart
- Probability distribution