Math 2.0

Statistics Definition

Consistent estimator

Builds on EstimatorConvergence in probability

Definition

For each n≥1n \ge 1, let θ^n\hat{\theta}_n be an estimator of a real parameter θ\theta that uses the first nn observations of a sample. The sequence (θ^n)(\hat{\theta}_n) is consistent if, for each θ∈Θ\theta \in \Theta, θ^n→θ\hat{\theta}_n \to \theta in probability when θ\theta is the true parameter: lim⁡n→∞Pθ(∣θ^n−θ∣>ε)=0for each ε>0.\lim_{n \to \infty} P_\theta\left(|\hat{\theta}_n - \theta| > \varepsilon\right) = 0 \quad \text{for each } \varepsilon > 0.

Remarks

If the convergence is almost sure, the sequence is strongly consistent. By the weak law of large numbers, the sample mean of an i.i.d. sample is a consistent estimator of the mean. If the mean squared error of θ^n\hat{\theta}_n tends to 00, then (θ^n)(\hat{\theta}_n) is consistent. This follows from Markov's inequality. Consistent estimators can be biased, and unbiased estimators can fail to be consistent. For example, the estimator X1X_1 of the mean uses only one observation: it is unbiased but not consistent, unless X1X_1 is almost surely constant.

Reading path

To understand this page from the ground up, read these 51 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. Convergence in probability definition
  44. Mutual independence of events definition
  45. Identically distributed random variables definition
  46. Independent sigma-algebras definition
  47. Independence of random variables definition
  48. Independent and identically distributed random variables definition
  49. Statistic definition
  50. Estimator definition
  51. Consistent estimator definition

Where this leads

19 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Consistent_estimator, Definition
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DefinesConsistent estimator
UsesEstimator, Convergence in probability

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