Math 2.0

Statistics Definition

Mathematical statistics

Builds on Statistical modelStatistical inferenceStatisticProbability theory

Definition

Mathematical statistics is the study of statistical inference with the methods of probability theory. Its setting is a statistical model (S,𝒫)(S, \mathcal{P}). The data x∈Sx \in S are a value of a random element XX whose law PP is an unknown member of 𝒫\mathcal{P}. Its objects are statistics T=g(X)T = g(X) and the procedures built from them. It judges each procedure by the laws that the procedure has under the members of 𝒫\mathcal{P}.

Remarks

Example: in the model of nn independent 𝒩(μ,1)\mathcal{N}(\mu, 1) observations with μ∈ℝ\mu \in \R, the sample mean Xˉn\bar{X}_n has law 𝒩(μ,1/n)\mathcal{N}(\mu, 1/n) for every μ\mu. This law shows that Xˉn\bar{X}_n is unbiased for μ\mu with variance 1/n1/n, a typical result of mathematical statistics. Non-example: drawing a bar chart of one data set is descriptive statistics; it uses no model 𝒫\mathcal{P} and makes no claim about other possible data. The main branches are point estimation, hypothesis testing, confidence sets and decision theory, each in a parametric or a nonparametric form.

Reading path

To understand this page from the ground up, read these 53 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. Probability theory concept
  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. Preimage of a set definition
  33. Sigma-algebra definition
  34. Event definition
  35. Borel sigma-algebra definition
  36. Probability axioms concept
  37. Probability measure definition
  38. Probability distribution definition
  39. Statistical model definition
  40. Probability space definition
  41. Conditional probability definition
  42. Random variable definition
  43. Law of a random variable definition
  44. Independence of two events definition
  45. Identically distributed random variables definition
  46. Mutual independence of events definition
  47. Independent sigma-algebras definition
  48. Independence of random variables definition
  49. Independent and identically distributed random variables definition
  50. Statistic definition
  51. Estimator definition
  52. Statistical inference definition
  53. Mathematical statistics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mathematical_statistics
About this page
Tag2W9FEY4Y6K
URIm2://2W9FEY4Y6K
Kinddefinition
WritingW0 stub
MSC62A01
DefinesMathematical statistics
UsesStatistical model, Statistical inference, Statistic, Probability theory

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