Math 2.0

Statistics Definition

P-value

Builds on Statistical hypothesis testStatisticContinuous uniform distribution

Definition

Consider a hypothesis test of a null hypothesis H0 ⁣:θ∈Θ0H_0 \colon \theta \in \Theta_0 with a test statistic T(X)T(X), where large values of TT are evidence against H0H_0. Let tt be the observed value of T(X)T(X). The p-value of the observation is p(t)=sup⁡θ∈Θ0Pθ(T(X)≥t).p(t) = \sup_{\theta \in \Theta_0} P_\theta\big(T(X) \ge t\big). If H0H_0 is simple, Θ0={θ0}\Theta_0 = \{\theta_0\}, then p(t)=Pθ0(T(X)≥t)p(t) = P_{\theta_0}(T(X) \ge t).

Remarks

For a two-sided test with a statistic that is symmetric about 00 under H0H_0, one uses Pθ0(∣T(X)∣≥∣t∣)P_{\theta_0}(|T(X)| \ge |t|). The test that rejects H0H_0 when p≤αp \le \alpha has significance level α\alpha. If H0H_0 is simple and T(X)T(X) has a continuous law under H0H_0, then the random p-value p(T(X))p(T(X)) has the uniform distribution on [0,1][0, 1] under H0H_0. The p-value is not the probability that H0H_0 is true.

Reading path

To understand this page from the ground up, read these 54 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. Cumulative distribution function definition
  43. Law of a random variable definition
  44. Random variable with a density definition
  45. Mutual independence of events definition
  46. Identically distributed random variables definition
  47. Probability density function definition
  48. Independent sigma-algebras definition
  49. Continuous uniform distribution definition
  50. Independence of random variables definition
  51. Independent and identically distributed random variables definition
  52. Statistic definition
  53. Statistical hypothesis test concept
  54. P-value definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/P-value, Definition
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UsesStatistical hypothesis test, Statistic, Continuous uniform distribution

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