Math 2.0

Statistics Definition

Frequentist inference

Builds on Statistical inferenceSampling distributionStatistical modelConfidence interval

Definition

Let {Pθ:θ∈Θ}\set{P_\theta : \theta \in \Theta} be a statistical model and let XX be the data, with law PθP_\theta for a fixed unknown θ\theta. Frequentist inference is the approach to statistical inference in which:

  1. the parameter θ\theta is treated as fixed, and no probability distribution is placed on Θ\Theta;
  2. a procedure δ(X)\delta(X) is assessed through its sampling distribution under PθP_\theta;
  3. a guarantee about δ\delta must hold for every θ∈Θ\theta \in \Theta, or be stated as a function of θ\theta.

Remarks

Example: a confidence set of level 1−α1 - \alpha is a set-valued statistic C(X)C(X) with Pθ(θ∈C(X))≥1−αP_\theta(\theta \in C(X)) \ge 1 - \alpha for every θ∈Θ\theta \in \Theta. For nn i.i.d. 𝒩(θ,1)\mathcal{N}(\theta, 1) observations, C(X)=[Xˉn−z/n,Xˉn+z/n]C(X) = [\bar{X}_n - z/\sqrt{n}, \bar{X}_n + z/\sqrt{n}], with zz the 1−α/21 - \alpha/2 quantile of 𝒩(0,1)\mathcal{N}(0, 1), has coverage exactly 1−α1 - \alpha for every θ\theta, since n(Xˉn−θ)∼𝒩(0,1)\sqrt{n}(\bar{X}_n - \theta) \sim \mathcal{N}(0, 1). Non-example: a Bayesian credible interval is computed from a posterior distribution on Θ\Theta; it violates clause 1, and in general it has no coverage guarantee that holds for every θ\theta. A test of level α\alpha is likewise a procedure whose probability of rejecting a true hypothesis is at most α\alpha under every θ\theta of the hypothesis.

Reading path

To understand this page from the ground up, read these 78 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. Statistical model definition
  43. Probability space definition
  44. Conditional probability definition
  45. Random variable definition
  46. Discrete random variable definition
  47. Cumulative distribution function definition
  48. Law of a random variable definition
  49. Standard score definition
  50. Independence of two events definition
  51. Random variable with a density definition
  52. Joint probability distribution definition
  53. Identically distributed random variables definition
  54. Expected value definition
  55. Independence of two random variables definition
  56. Mutual independence of events definition
  57. Probability density function definition
  58. Marginal distribution definition
  59. Variance definition
  60. Linearity of expectation statement
  61. Independent sigma-algebras definition
  62. Density of the normal distribution definition
  63. Independence means the joint law is the product of the marginals statement
  64. Covariance definition
  65. Independence of random variables definition
  66. Normal distribution definition
  67. Variance of a sum statement
  68. Independent and identically distributed random variables definition
  69. Expectation of a product of independent random variables statement
  70. Statistic definition
  71. Independent random variables are uncorrelated statement
  72. Estimator definition
  73. Sampling distribution definition
  74. Variance of a sum of independent random variables statement
  75. Statistical inference definition
  76. Sample mean definition
  77. Confidence interval definition
  78. Frequentist inference definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Frequentist_inference
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TagYNBTV2S7SR
URIm2://YNBTV2S7SR
Kinddefinition
WritingW0 stub
MSC62A01
DefinesFrequentist inference
UsesStatistical inference, Sampling distribution, Statistical model, Confidence interval

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