Math 2.0

Statistics Definition

Wishart distribution

Builds on Multivariate normal distributionIndependent and identically distributed random variablesPositive definite matrixChi-squared distributionDeterminantTrace of a matrix

Definition

Let pp and nn be positive integers and let Σ\Sigma be a p×pp \times p positive definite matrix. Let X1,…,XnX_1, \dots, X_n be independent and identically distributed random column vectors in ℝp\R^p with the multivariate normal distribution 𝒩p(0,Σ)\mathcal{N}_p(0, \Sigma). The distribution of the random p×pp \times p matrix S=∑i=1nXiXiTS = \sum_{i=1}^n X_i X_i^T is the Wishart distribution with scale matrix Σ\Sigma and nn degrees of freedom, written Wp(Σ,n)W_p(\Sigma, n).

Remarks

The mean is E(S)=nΣE(S) = n \Sigma. If n≥pn \ge p, then SS is positive definite with probability 11. Its density on positive definite matrices is then f(S)=det⁡(S)(n−p−1)/2exp⁡(−12tr⁡(Σ−1S))2np/2det⁡(Σ)n/2 Γp(n/2)f(S) = \frac{\det(S)^{(n - p - 1)/2} \exp\left(-\tfrac{1}{2} \tr(\Sigma^{-1} S)\right)}{2^{np/2} \det(\Sigma)^{n/2} \, \Gamma_p(n/2)} where det⁡\det is the determinant, tr⁡\tr is the trace and Γp\Gamma_p is the multivariate gamma function.

Example: for p=1p = 1 and Σ=(σ2)\Sigma = (\sigma^2), S=∑iXi2=σ2∑i(Xi/σ)2S = \sum_i X_i^2 = \sigma^2 \sum_i (X_i/\sigma)^2 with Xi/σX_i/\sigma standard normal, so W1(σ2,n)W_1(\sigma^2, n) is σ2\sigma^2 times the chi-squared distribution with nn degrees of freedom.

Example: if Y1,…,YmY_1, \dots, Y_m are i.i.d. 𝒩p(μ,Σ)\mathcal{N}_p(\mu, \Sigma) with m≥2m \ge 2 and Yˉ\bar{Y} is their mean, then ∑i=1m(Yi−Yˉ)(Yi−Yˉ)T∼Wp(Σ,m−1)\sum_{i=1}^m (Y_i - \bar{Y})(Y_i - \bar{Y})^T \sim W_p(\Sigma, m - 1).

Non-example: if the XiX_i have a common mean μ≠0\mu \ne 0, then E(S)=nΣ+nμμT≠nΣE(S) = n\Sigma + n\mu\mu^T \ne n\Sigma, so SS is not Wp(Σ,n)W_p(\Sigma, n); its law is a noncentral Wishart distribution.

Reading path

To understand this page from the ground up, read these 93 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Identity matrix object
  4. Integers object
  5. Matrix definition
  6. Ordered field definition
  7. Proposition concept
  8. Random experiment concept
  9. Rational numbers object
  10. Real numbers object
  11. Vector space definition
  12. Set definition
  13. Ring of integers statement
  14. Square matrix definition
  15. Transpose definition
  16. Disjunction definition
  17. Implication definition
  18. Predicate definition
  19. Linear map definition
  20. Ordered pair definition
  21. Union definition
  22. Trace of a matrix definition
  23. Symmetric matrix definition
  24. Conjugate transpose definition
  25. Sample space definition
  26. The rationals form a field statement
  27. Existential quantifier definition
  28. Universal quantifier definition
  29. Multilinear map definition
  30. Cartesian product definition
  31. Hermitian matrix definition
  32. Subset definition
  33. The rationals form an ordered field statement
  34. Uniqueness quantifier definition
  35. Alternating multilinear map definition
  36. Binary relation definition
  37. Universal set definition
  38. Order on the reals definition
  39. Positive definite matrix definition
  40. Power set definition
  41. Determinant definition
  42. Complement definition
  43. Function definition
  44. Interval definition
  45. Injective function definition
  46. Sigma-algebra definition
  47. Indicator function definition
  48. Preimage of a set definition
  49. Countable set definition
  50. Event definition
  51. Borel sigma-algebra definition
  52. Probability axioms concept
  53. Probability measure definition
  54. Dirac measure definition
  55. Probability distribution definition
  56. Probability space definition
  57. Conditional probability definition
  58. Random variable definition
  59. Discrete random variable definition
  60. Cumulative distribution function definition
  61. Independence of two events definition
  62. Law of a random variable definition
  63. Random variable with a density definition
  64. Joint probability distribution definition
  65. Independence of two random variables definition
  66. Mutual independence of events definition
  67. Expected value definition
  68. Probability density function definition
  69. Identically distributed random variables definition
  70. Independent sigma-algebras definition
  71. Marginal distribution definition
  72. Variance definition
  73. Linearity of expectation statement
  74. Density of the normal distribution definition
  75. Exponential distribution definition
  76. Independence of random variables definition
  77. Independence means the joint law is the product of the marginals statement
  78. Covariance definition
  79. Normal distribution definition
  80. Gamma distribution definition
  81. Expectation of a product of independent random variables statement
  82. Multivariate normal distribution definition
  83. Variance of a sum statement
  84. Independent and identically distributed random variables definition
  85. Independent random variables are uncorrelated statement
  86. Statistic definition
  87. Variance of a sum of independent random variables statement
  88. Estimator definition
  89. Sample mean definition
  90. Bias of an estimator definition
  91. Sample variance definition
  92. Chi-squared distribution definition
  93. Wishart distribution definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wishart_distribution
About this page
TagFRNDCF91FT
URIm2://FRNDCF91FT
Kinddefinition
WritingW0 stub
MSC62H10, 60E05
DefinesWishart distribution
UsesMultivariate normal distribution, Independent and identically distributed random variables, Positive definite matrix, Chi-squared distribution, Determinant, Trace of a matrix

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