Math 2.0

Statistics Definition

Wald test

Builds on Maximum likelihood estimationFisher informationStatistical hypothesis testChi-squared distribution

Definition

Let XX be data from a model with parameter θ\theta in an open set Θ⊆ℝk\Theta \subseteq \R^k, and let θ0∈Θ\theta_0 \in \Theta. Let θ^\hat\theta be the maximum likelihood estimate of θ\theta, and let I(θ^)I(\hat\theta) be the Fisher information of XX evaluated at θ^\hat\theta, assumed invertible. The Wald statistic for H0 ⁣:θ=θ0H_0 \colon \theta = \theta_0 is W=(θ^−θ0)TI(θ^)(θ^−θ0).W = (\hat\theta - \theta_0)^{T} I(\hat\theta) (\hat\theta - \theta_0). The Wald test at level α\alpha rejects H0H_0 when WW exceeds the 1−α1 - \alpha quantile of the chi-squared distribution with kk degrees of freedom.

Remarks

Under H0H_0 and the usual regularity conditions for i.i.d. samples, WW converges in distribution to χk2\chi^2_k, so the test has asymptotic level α\alpha. For k=1k = 1 the test rejects when ∣θ^−θ0∣/se^>z1−α/2\abs{\hat\theta - \theta_0}/\widehat{\mathrm{se}} > z_{1 - \alpha/2}, with se^=I(θ^)−1/2\widehat{\mathrm{se}} = I(\hat\theta)^{-1/2} and z1−α/2z_{1 - \alpha/2} a standard normal quantile. The observed information may replace I(θ^)I(\hat\theta) without changing the limit law. Example: for XX binomial with parameters nn and pp, one has p^=X/n\hat p = X/n and W=(p^−p0)2n/(p^(1−p^))W = (\hat p - p_0)^2 n/(\hat p(1 - \hat p)). Unlike the likelihood-ratio test, the Wald test is not invariant under a reparametrization of θ\theta.

Reading path

To understand this page from the ground up, read these 85 pages in order. The foundations come first.

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Wald_test
About this page
TagT9640YF4TX
URIm2://T9640YF4TX
Kinddefinition
WritingW0 stub
MSC62F03, 62F05
DefinesWald test
UsesMaximum likelihood estimation, Fisher information, Statistical hypothesis test, Chi-squared distribution

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