Math 2.0

Statistics Definition

Sampling distribution

Builds on Statistical modelStatisticProbability distribution

Definition

Let 𝒫={Pθ:θ∈Θ}\mathcal{P} = \set{P_\theta : \theta \in \Theta} be a statistical model on a measurable space (S,𝒮)(S, \mathcal{S}), and let XX be a random element of SS. Let T=g(X)T = g(X) be a statistic with values in ℝk\R^k. The sampling distribution of TT under θ\theta is the probability distribution of TT when XX has law PθP_\theta: Pθ(T∈B)=Pθ(g−1(B))P_\theta(T \in B) = P_\theta\left(g^{-1}(B)\right) for every Borel set B⊆ℝkB \subseteq \R^k.

Remarks

Example: let X1,…,XnX_1, \dots, X_n be i.i.d. 𝒩(μ,σ2)\mathcal{N}(\mu, \sigma^2). The sample mean Xˉn\bar{X}_n is a linear combination of independent normal variables, so its sampling distribution is 𝒩(μ,σ2/n)\mathcal{N}(\mu, \sigma^2/n). Non-example: the histogram of one observed data set is not the sampling distribution of anything; it is a single realization, while a sampling distribution ranges over all possible data. The standard deviation of the sampling distribution of an estimator is its standard error. In general the sampling distribution depends on θ\theta; a statistic whose sampling distribution is the same for all θ\theta is ancillary.

Reading path

To understand this page from the ground up, read these 50 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. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Statistical model definition
  39. Probability space definition
  40. Conditional probability definition
  41. Random variable definition
  42. Law of a random variable definition
  43. Independence of two events definition
  44. Identically distributed random variables definition
  45. Mutual independence of events definition
  46. Independent sigma-algebras definition
  47. Independence of random variables definition
  48. Independent and identically distributed random variables definition
  49. Statistic definition
  50. Sampling distribution definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Sampling_distribution
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URIm2://FC9NRV5GSP
Kinddefinition
WritingW0 stub
MSC62E15, 62A01
DefinesSampling distribution
UsesStatistical model, Statistic, Probability distribution

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