theorem 11.67 Asymptotics of the MLE

open in the book · parts/02-mathematical-methods/09-probability-statistics.tex:1934 · p. 394

Rests on

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 definition 11.4 Expectation ¶
      - depends_on proposition 11.2 Elementary rules ¶
      - proves proof ch:09-probability-statistics@proof-2 ¶
  - 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 11.54 Likelihood ¶
  - depends_on definition 11.53 Statistical model ¶
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 11.61 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 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 11.46 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 ¶

Supports

depends_on proposition 11.72 Nuisance parameters cost information ¶
depends_on theorem 11.88 Wilks ¶
- 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`	→	Weak law of large numbers	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	→	Maximum-likelihood estimator	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	→	Regularity conditions	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	→	Score identities	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	→	Slutsky	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	→	Central limit theorem, Lindeberg–Lévy	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1949
`depends_on`	←	Nuisance parameters cost information	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2253
`depends_on`	←	Wilks	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2720
`proves`	←	ch:09-probability-statistics@proof-27	declared	parts/02-mathematical-methods/09-probability-statistics.tex:1952
`proves`	←	ch:09-probability-statistics@proof-29	declared	parts/02-mathematical-methods/09-probability-statistics.tex:2088