Math 2.0

Statistics Definition

Mean squared error

Builds on EstimatorBias of an estimatorVarianceExpected value

Definition

Let θ^\hat{\theta} be an estimator of a real parameter θ\theta, with Eθ[θ^2]<∞E_\theta[\hat{\theta}^2] < \infty, where EθE_\theta is the expectation when θ\theta is the true parameter. The mean squared error of θ^\hat{\theta} at θ\theta is MSE⁡θ(θ^)=Eθ[(θ^−θ)2].\operatorname{MSE}_\theta(\hat{\theta}) = E_\theta\left[(\hat{\theta} - \theta)^2\right].

Remarks

It decomposes as MSE⁡θ(θ^)=Var⁡θ(θ^)+Bias⁡θ(θ^)2\operatorname{MSE}_\theta(\hat{\theta}) = \operatorname{Var}_\theta(\hat{\theta}) + \operatorname{Bias}_\theta(\hat{\theta})^2, with the bias Bias⁡θ(θ^)=Eθ[θ^]−θ\operatorname{Bias}_\theta(\hat{\theta}) = E_\theta[\hat{\theta}] - \theta. For an unbiased estimator, the mean squared error is the variance. For the sample mean of nn i.i.d. variables with variance σ2\sigma^2, the mean squared error is σ2/n\sigma^2/n. For nn predictions y^i\hat{y}_i of observed values yiy_i, one also calls 1n∑i(yi−y^i)2\frac{1}{n}\sum_i (y_i - \hat{y}_i)^2 the mean squared error.

Reading path

To understand this page from the ground up, read these 57 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. Variance definition
  51. Independent sigma-algebras definition
  52. Independence of random variables definition
  53. Independent and identically distributed random variables definition
  54. Statistic definition
  55. Estimator definition
  56. Bias of an estimator definition
  57. Mean squared error definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Mean_squared_error, Estimator
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DefinesMean squared error
UsesEstimator, Bias of an estimator, Variance, Expected value

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