Math 2.0

Statistics Definition

Estimator

Builds on StatisticIndependent and identically distributed random variables

Definition

Let Θ\Theta be a set of parameters, and for each θ∈Θ\theta \in \Theta let PθP_\theta be a probability distribution. Let X1,…,XnX_1, \dots, X_n be a sample whose joint law is determined by an unknown θ∈Θ\theta \in \Theta. Let τ ⁣:Θ→ℝk\tau \colon \Theta \to \mathbb{R}^k be a function. An estimator of τ(θ)\tau(\theta) is a statistic τ^=g(X1,…,Xn)\hat{\tau} = g(X_1, \dots, X_n) with values in ℝk\mathbb{R}^k, where gg does not depend on θ\theta. The value of τ^\hat{\tau} for observed data is an estimate of τ(θ)\tau(\theta).

Remarks

The definition does not ask that τ^\hat{\tau} is close to τ(θ)\tau(\theta). One measures the quality of an estimator with its bias, its variance and its mean squared error. For an i.i.d. sample, the sample mean is an estimator of the mean, and the sample variance is an estimator of the variance. One often writes θ^\hat{\theta} for an estimator of θ\theta itself.

Reading path

To understand this page from the ground up, read these 49 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. Probability space definition
  39. Conditional probability definition
  40. Random variable definition
  41. Independence of two events definition
  42. Law of a random variable definition
  43. Mutual independence of events definition
  44. Identically distributed random variables definition
  45. Independent sigma-algebras definition
  46. Independence of random variables definition
  47. Independent and identically distributed random variables definition
  48. Statistic definition
  49. Estimator definition

Where this leads

53 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Estimator, Definition
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DefinesEstimator
UsesStatistic, Independent and identically distributed random variables

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