theorem 11.88 Wilks

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:2708 · p. 403

Rests on

depends_on definition 11.80 Chi-squared ¶
depends_on definition 11.87 Likelihood-ratio test statistic ¶
- depends_on definition 11.54 Likelihood ¶
  - depends_on definition 11.53 Statistical model ¶
- depends_on definition 11.56 Maximum-likelihood estimator ¶
  - depends_on definition 11.54 Likelihood ¶ ↺
- depends_on definition 11.70 Profile likelihood ¶
  - depends_on definition 11.54 Likelihood ¶ ↺
  - depends_on definition 11.56 Maximum-likelihood estimator ¶ ↺
depends_on definition 11.59 Regularity conditions ¶
- depends_on definition 11.60 Score and Fisher information ¶
  - depends_on definition 11.53 Statistical model ¶ ↺
- depends_on definition 11.53 Statistical model ¶ ↺
depends_on lemma 11.83 Gaussian quadratic forms ¶
- depends_on definition 11.80 Chi-squared ¶ ↺
- proves proof ch:09-probability-statistics@proof-36 ¶
depends_on lemma 11.52 Slutsky ¶
- depends_on definition 11.21 Distribution function ¶
  - depends_on definition 11.20 Density ¶
    - depends_on definition 7.39 Darboux sums and the definite integral ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
- proves proof ch:09-probability-statistics@proof-20 ¶
- proves proof ch:09-probability-statistics@proof-21 ¶
depends_on theorem 11.67 Asymptotics of the MLE ¶
- depends_on corollary 11.51 Weak law of large numbers ¶
  - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
    - depends_on definition 11.6 Moments, variance, standard deviation ¶
      - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶
    - depends_on proposition 11.5 Transfer, linearity, monotonicity ¶ ↺
    - proves proof ch:09-probability-statistics@proof-18 ¶
  - proves proof ch:09-probability-statistics@proof-19 ¶
- depends_on definition 11.56 Maximum-likelihood estimator ¶ ↺
- depends_on definition 11.59 Regularity conditions ¶ ↺
- depends_on lemma 11.61 Score identities ¶
  - depends_on definition 11.60 Score and Fisher information ¶ ↺
  - depends_on definition 11.59 Regularity conditions ¶ ↺
  - proves proof ch:09-probability-statistics@proof-23 ¶
- depends_on lemma 11.52 Slutsky ¶ ↺
- depends_on theorem 11.46 Central limit theorem, Lindeberg–Lévy ¶
  - depends_on lemma 11.44 Second-order expansion ¶
    - depends_on definition 11.42 Characteristic function ¶
    - proves proof ch:09-probability-statistics@proof-16 ¶
  - depends_on proposition 11.43 Elementary properties ¶
    - depends_on definition 11.42 Characteristic function ¶ ↺
    - proves proof ch:09-probability-statistics@proof-15 ¶
  - depends_on theorem 11.45 Lévy's continuity theorem ¶
    - depends_on definition 11.42 Characteristic function ¶ ↺
    - proves proof ch:09-probability-statistics@prooflink-1 ¶
  - proves proof ch:09-probability-statistics@proof-17 ¶
- proves proof ch:09-probability-statistics@proof-27 ¶
- proves proof ch:09-probability-statistics@proof-29 ¶
proves proof ch:09-probability-statistics@proof-38 ¶
proves proof ch:09-probability-statistics@proof-39 ¶
proves proof ch:09-probability-statistics@prooflink-4 ¶

Supports

depends_on corollary 11.89 Degrees of freedom with nuisance parameters ¶
- depends_on example 11.107 Degrading a local five sigma ¶
- depends_on proposition 11.93 $Z=\sqrt{q}$ ¶
  - depends_on example 11.107 Degrading a local five sigma ¶ ↺
  - depends_on proposition 113.15 Why categorizing helps, and by how much ¶
  - depends_on proposition 113.12 Sensitivity is set by ratio, not by rate ¶
- depends_on remark 11.90 Failure I: a parameter on the boundary ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Chi-squared	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	→	Likelihood-ratio test statistic	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	→	Regularity conditions	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	→	Gaussian quadratic forms	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	→	Slutsky	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	→	Asymptotics of the MLE	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`depends_on`	←	Degrees of freedom with nuisance parameters	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2815
`proves`	←	ch:09-probability-statistics@proof-38	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2723
`proves`	←	ch:09-probability-statistics@proof-39	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2765
`proves`	←	ch:09-probability-statistics@prooflink-4	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2806