Math 2.0

Statistics Definition

Neyman construction

Builds on Confidence intervalStatistical hypothesis test

Definition

Let XX be data with values in a set 𝒳\mathcal{X}, whose law PθP_\theta depends on a parameter θ∈Θ\theta \in \Theta, and let α∈(0,1)\alpha \in (0, 1). For each θ∈Θ\theta \in \Theta, choose an acceptance region A(θ)⊆𝒳A(\theta) \subseteq \mathcal{X} with Pθ(X∈A(θ))≥1−αP_\theta(X \in A(\theta)) \ge 1 - \alpha. The Neyman construction is the set-valued statistic C(X)={θ∈Θ:X∈A(θ)}.C(X) = \set{\theta \in \Theta : X \in A(\theta)}.

Remarks

For each θ\theta, the events {θ∈C(X)}\set{\theta \in C(X)} and {X∈A(θ)}\set{X \in A(\theta)} are equal, so Pθ(θ∈C(X))≥1−αP_\theta(\theta \in C(X)) \ge 1 - \alpha. So C(X)C(X) is a confidence set of level 1−α1 - \alpha; its coverage is exactly 1−α1 - \alpha when each A(θ)A(\theta) has probability exactly 1−α1 - \alpha. Equivalently, A(θ)A(\theta) is the acceptance region of a test at level α\alpha of the hypothesis that θ\theta is the true value. Then C(X)C(X) is the set of values that these tests do not reject. Example: for X∼𝒩(θ,1)X \sim \mathcal{N}(\theta, 1) and A(θ)=[θ−1.96,θ+1.96]A(\theta) = [\theta - 1.96, \theta + 1.96], the construction gives C(X)=[X−1.96,X+1.96]C(X) = [X - 1.96, X + 1.96], of level about 0.950.95. Non-example: for α=0.05\alpha = 0.05, the choice A(θ)=[θ−1,θ+1]A(\theta) = [\theta - 1, \theta + 1] fails the condition, since Pθ(X∈A(θ))≈0.68P_\theta(X \in A(\theta)) \approx 0.68; the resulting set has level only about 0.680.68.

Reading path

To understand this page from the ground up, read these 75 pages in order. The foundations come first.

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

Further reading

  1. Web https://en.wikipedia.org/wiki/Neyman_construction
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WritingW0 stub
MSC62F25, 62F03
DefinesNeyman construction
UsesConfidence interval, Statistical hypothesis test

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