lemma A.213 The three minima

open in the book · appendices/A-long-proofs.tex:10786 · p. 2895

Rests on

depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
- depends_on lemma A.208 The three averages ¶
  - depends_on lemma 11.61 Score identities ¶
    - depends_on definition 11.60 Score and Fisher information ¶
      - 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 ¶ ↺
    - proves proof ch:09-probability-statistics@proof-23 ¶
  - depends_on theorem A.206 Cramér–Wold, and continuous mapping ¶
    - depends_on definition 11.21 Distribution function ¶
      - depends_on definition 11.20 Density ¶
    - depends_on lemma 11.52 Slutsky ¶
      - depends_on definition 11.21 Distribution function ¶ ↺
      - proves proof ch:09-probability-statistics@proof-20 ¶
      - proves proof ch:09-probability-statistics@proof-21 ¶
  - depends_on theorem A.205 Weak law under integrability alone ¶
    - depends_on corollary 11.51 Weak law of large numbers ¶
      - depends_on lemma 11.50 Markov and Chebyshev inequalities ¶
      - proves proof ch:09-probability-statistics@proof-19 ¶
  - proves proof app:A-long-proofs@proof-126 ¶
- depends_on lemma A.209 Uniform third-order remainder ¶
  - depends_on definition 11.59 Regularity conditions ¶ ↺
  - depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
    - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - proves proof ch:05-real-analysis@proof-14 ¶
    - depends_on theorem 7.34 Rolle ¶
      - depends_on lemma 7.33 Fermat: interior extremum ¶
      - depends_on theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - proves proof ch:05-real-analysis@proof-22 ¶
  - proves proof app:A-long-proofs@proof-127 ¶
- proves proof app:A-long-proofs@proof-128 ¶
depends_on lemma A.212 Second-order flatness ¶
- depends_on definition A.203 The constraint surface ¶
  - 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 theorem 7.38 Taylor's theorem with Lagrange remainder ¶ ↺
- proves proof app:A-long-proofs@proof-130 ¶
depends_on lemma A.211 Root-$n$ localisation ¶
- depends_on lemma A.208 The three averages ¶ ↺
- depends_on lemma A.209 Uniform third-order remainder ¶ ↺
- proves proof app:A-long-proofs@proof-129 ¶
proves proof app:A-long-proofs@proof-131 ¶

Supports

depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶

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`	→	Uniform quadratic approximation on the $n^{-1/2}$ scale	declared	appendices/A-long-proofs.tex:10799
`depends_on`	→	Second-order flatness	declared	appendices/A-long-proofs.tex:10799
`depends_on`	→	Root-$n$ localisation	declared	appendices/A-long-proofs.tex:10799
`depends_on`	←	Wilks, $k$ parameters and $r$ constraints	declared	appendices/A-long-proofs.tex:10914
`proves`	←	app:A-long-proofs@proof-131	declared	appendices/A-long-proofs.tex:10803