Math 2.0

Statistics Definition

Score test

Builds on ScoreFisher informationStatistical hypothesis testChi-squared distribution

Definition

Let XX be data with log-likelihood ℓ(θ)=log⁡L(θ∣X)\ell(\theta) = \log L(\theta \mid X), differentiable in θ\theta in an open set Θ⊆ℝk\Theta \subseteq \R^k, and let θ0∈Θ\theta_0 \in \Theta. Let U(θ)=∇ℓ(θ)U(\theta) = \nabla \ell(\theta) be the score and let I(θ)I(\theta) be the Fisher information of XX, assumed invertible at θ0\theta_0. The score statistic for H0 ⁣:θ=θ0H_0 \colon \theta = \theta_0 is S=U(θ0)TI(θ0)−1U(θ0).S = U(\theta_0)^{T} I(\theta_0)^{-1} U(\theta_0). The score test at level α\alpha rejects H0H_0 when SS 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, SS converges in distribution to χk2\chi^2_k, so the test has asymptotic level α\alpha. For a composite H0H_0 with nuisance parameters, one evaluates the score and the information at the maximum likelihood estimate under H0H_0. Example: for XX binomial with parameters nn and pp, one has U(p)=(X−np)/(p(1−p))U(p) = (X - np)/(p(1 - p)) and I(p)=n/(p(1−p))I(p) = n/(p(1 - p)). So S=(X−np0)2/(np0(1−p0))S = (X - np_0)^2/(np_0(1 - p_0)), the square of the usual z-statistic for a proportion. Non-example: the statistic U(θ0)TU(θ0)U(\theta_0)^T U(\theta_0), without the factor I(θ0)−1I(\theta_0)^{-1}, does not have a χk2\chi^2_k limit in general, so it is not the score statistic.

Reading path

To understand this page from the ground up, read these 81 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. Score concept
  11. Set definition
  12. Ring of integers statement
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. Existential quantifier definition
  20. The rationals form a field statement
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Order on the reals definition
  29. Power set definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Indicator function definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Sigma-algebra definition
  37. Countable set definition
  38. Event definition
  39. Borel sigma-algebra definition
  40. Probability axioms concept
  41. Probability measure definition
  42. Dirac measure definition
  43. Probability distribution definition
  44. Probability space definition
  45. Conditional probability definition
  46. Random variable definition
  47. Discrete random variable 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. Joint probability distribution definition
  53. Independence of two random variables definition
  54. Mutual independence of events definition
  55. Expected value definition
  56. Probability density function definition
  57. Identically distributed random variables definition
  58. Marginal distribution definition
  59. Variance definition
  60. Independent sigma-algebras definition
  61. Linearity of expectation statement
  62. Density of the normal distribution definition
  63. Exponential distribution definition
  64. Independence means the joint law is the product of the marginals statement
  65. Independence of random variables definition
  66. Covariance definition
  67. Normal distribution definition
  68. Gamma distribution definition
  69. Expectation of a product of independent random variables statement
  70. Variance of a sum statement
  71. Independent and identically distributed random variables definition
  72. Independent random variables are uncorrelated statement
  73. Statistic definition
  74. Variance of a sum of independent random variables statement
  75. Statistical hypothesis test concept
  76. Estimator definition
  77. Sample mean definition
  78. Bias of an estimator definition
  79. Sample variance definition
  80. Chi-squared distribution definition
  81. Score test definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Score_test
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MSC62F03, 62F05
DefinesScore test
UsesScore, Fisher information, Statistical hypothesis test, Chi-squared distribution

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