Math 2.0

Statistics Definition

Bias of an estimator

Builds on EstimatorExpected value

Definition

Let θ^\hat{\theta} be an estimator of a real parameter θ\theta, with Eθ[∣θ^∣]<∞E_\theta[|\hat{\theta}|] < \infty for each θ\theta. Here EθE_\theta is the expectation when θ\theta is the true parameter. The bias of θ^\hat{\theta} at θ\theta is Bias⁡θ(θ^)=Eθ[θ^]−θ.\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta. The estimator is unbiased if Bias⁡θ(θ^)=0\operatorname{Bias}_\theta(\hat{\theta}) = 0 for each θ\theta.

Remarks

For an i.i.d. sample with mean μ\mu, the sample mean is an unbiased estimator of μ\mu. The sample variance with divisor nn has bias −σ2/n-\sigma^2/n. With divisor n−1n - 1, it is unbiased. An unbiased estimator need not be good. The mean squared error is variance plus squared bias, and a biased estimator can have a smaller mean squared error. Unbiasedness is not preserved by nonlinear maps. For example, the sample standard deviation SnS_n is a biased estimator of σ\sigma, even though Sn2S_n^2 is unbiased for σ2\sigma^2.

Reading path

To understand this page from the ground up, read these 55 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. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability distribution definition
  40. Probability space definition
  41. Conditional probability definition
  42. Random variable definition
  43. Discrete random variable definition
  44. Law of a random variable definition
  45. Independence of two events definition
  46. Random variable with a density definition
  47. Identically distributed random variables definition
  48. Expected value definition
  49. Mutual independence of events definition
  50. Independent sigma-algebras definition
  51. Independence of random variables definition
  52. Independent and identically distributed random variables definition
  53. Statistic definition
  54. Estimator definition
  55. Bias of an estimator definition

Where this leads

23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Bias_of_an_estimator, Definition
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