Math 2.0

Statistics Definition

Type I and type II errors

Builds on Statistical hypothesis test

Definition

Consider a hypothesis test of H0 ⁣:θ∈Θ0H_0 \colon \theta \in \Theta_0 against H1 ⁣:θ∈Θ1H_1 \colon \theta \in \Theta_1 that rejects H0H_0 when the data XX are in a rejection region RR. A type I error occurs when H0H_0 is true and the test rejects H0H_0. Its probability at θ∈Θ0\theta \in \Theta_0 is Pθ(X∈R)P_\theta(X \in R). A type II error occurs when H0H_0 is false and the test does not reject H0H_0. Its probability at θ∈Θ1\theta \in \Theta_1 is β(θ)=Pθ(X∉R)\beta(\theta) = P_\theta(X \notin R).

Remarks

The size of the test is α=sup⁡θ∈Θ0Pθ(X∈R)\alpha = \sup_{\theta \in \Theta_0} P_\theta(X \in R). The power of the test at θ∈Θ1\theta \in \Theta_1 is 1−β(θ)1 - \beta(\theta). For a fixed sample size, a smaller rejection region lowers the probability of a type I error. It raises the probability of a type II error. In the frequentist practice, one fixes a bound α\alpha on the type I error, and then chooses a test with a small type II error.

Reading path

To understand this page from the ground up, read these 50 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. Statistical hypothesis test concept
  50. Type I and type II errors definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Type_I_and_type_II_errors, Definition
  2. Web https://en.wikipedia.org/wiki/Type_I_and_type_II_errors, Error rate
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DefinesType I and type II errors
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