Math 2.0

Statistics Definition

Hotelling's T-squared distribution

Builds on Multivariate normal distributionWishart distributionIndependence of two random variablesPositive definite matrixF-distributionStudent's t-distribution

Definition

Let pp and mm be positive integers with m≥pm \ge p, and let Σ\Sigma be a p×pp \times p positive definite matrix. Let xx be a random vector with the multivariate normal distribution 𝒩p(0,Σ)\mathcal{N}_p(0, \Sigma). Let MM be a random matrix with the Wishart distribution Wp(Σ,m)W_p(\Sigma, m), independent of xx. The distribution of T2=m xTM−1xT^2 = m \, x^T M^{-1} x does not depend on Σ\Sigma. It is Hotelling's T-squared distribution with dimension pp and mm degrees of freedom, written T2(p,m)T^2(p, m).

Remarks

The condition m≥pm \ge p makes MM invertible with probability 11. The distribution is a rescaled F-distribution: m−p+1pmT2∼F(p,m−p+1).\frac{m - p + 1}{p m} T^2 \sim F(p, m - p + 1).

Example: for p=1p = 1, x=σZx = \sigma Z with ZZ standard normal and M=σ2VM = \sigma^2 V with VV chi-squared with mm degrees of freedom. Then T2=Z2/(V/m)T^2 = Z^2/(V/m) is the square of a Student t variable with mm degrees of freedom, and the formula above gives T2∼F(1,m)T^2 \sim F(1, m).

Example: let Y1,…,YnY_1, \dots, Y_n be i.i.d. 𝒩p(μ,Σ)\mathcal{N}_p(\mu, \Sigma) with n−1≥pn - 1 \ge p, with sample mean Yˉ\bar{Y} and sample covariance matrix Σ^\hat{\Sigma}. Then n(Yˉ−μ)∼𝒩p(0,Σ)\sqrt{n}(\bar{Y} - \mu) \sim \mathcal{N}_p(0, \Sigma) and (n−1)Σ^∼Wp(Σ,n−1)(n - 1)\hat{\Sigma} \sim W_p(\Sigma, n - 1) are independent, so n(Yˉ−μ)TΣ^−1(Yˉ−μ)∼T2(p,n−1)n (\bar{Y} - \mu)^T \hat{\Sigma}^{-1} (\bar{Y} - \mu) \sim T^2(p, n - 1).

Non-example: if MM and xx are dependent, for instance p=1p = 1, m=1m = 1 and M=x2M = x^2, then T2=1T^2 = 1 always, which is not T2(1,1)T^2(1, 1).

Reading path

To understand this page from the ground up, read these 90 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. F-distribution concept
  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. Wishart distribution concept
  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. Ordered pair definition
  20. Union definition
  21. Symmetric matrix definition
  22. Conjugate transpose definition
  23. Sample space definition
  24. The rationals form a field statement
  25. Existential quantifier definition
  26. Universal quantifier definition
  27. Cartesian product definition
  28. Hermitian matrix definition
  29. Subset definition
  30. The rationals form an ordered field statement
  31. Uniqueness quantifier definition
  32. Binary relation definition
  33. Universal set definition
  34. Order on the reals definition
  35. Positive definite matrix definition
  36. Power set definition
  37. Complement definition
  38. Function definition
  39. Interval definition
  40. Injective function definition
  41. Sigma-algebra definition
  42. Indicator function definition
  43. Preimage of a set definition
  44. Countable set definition
  45. Event definition
  46. Borel sigma-algebra definition
  47. Probability axioms concept
  48. Probability measure definition
  49. Dirac measure definition
  50. Probability distribution definition
  51. Probability space definition
  52. Conditional probability definition
  53. Random variable definition
  54. Discrete random variable definition
  55. Cumulative distribution function definition
  56. Independence of two events definition
  57. Law of a random variable definition
  58. Random variable with a density definition
  59. Joint probability distribution definition
  60. Independence of two random variables definition
  61. Mutual independence of events definition
  62. Expected value definition
  63. Probability density function definition
  64. Identically distributed random variables definition
  65. Independent sigma-algebras definition
  66. Marginal distribution definition
  67. Variance definition
  68. Linearity of expectation statement
  69. Cauchy distribution definition
  70. Density of the normal distribution definition
  71. Exponential distribution definition
  72. Independence of random variables definition
  73. Independence means the joint law is the product of the marginals statement
  74. Covariance definition
  75. Normal distribution definition
  76. Gamma distribution definition
  77. Expectation of a product of independent random variables statement
  78. Multivariate normal distribution definition
  79. Variance of a sum statement
  80. Independent and identically distributed random variables definition
  81. Independent random variables are uncorrelated statement
  82. Statistic definition
  83. Variance of a sum of independent random variables statement
  84. Estimator definition
  85. Sample mean definition
  86. Bias of an estimator definition
  87. Sample variance definition
  88. Chi-squared distribution definition
  89. Student's t-distribution definition
  90. Hotelling's T-squared distribution definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hotelling%27s_T-squared_distribution
About this page
TagSR8DQDKHQN
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Kinddefinition
WritingW0 stub
MSC62H10, 62H15
DefinesHotelling's T-squared distribution
UsesMultivariate normal distribution, Wishart distribution, Independence of two random variables, Positive definite matrix, F-distribution, Student's t-distribution

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