lemma A.209 Uniform third-order remainder

open in the book · appendices/A-long-proofs.tex:10551 · p. 2893

Rests on

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 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 definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - 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 definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-22 ¶
proves proof app:A-long-proofs@proof-127 ¶

Supports

depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
- depends_on lemma A.213 The three minima ¶
  - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
depends_on lemma A.211 Root-$n$ localisation ¶
- depends_on lemma A.213 The three minima ¶ ↺

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`	→	Regularity conditions	declared	appendices/A-long-proofs.tex:10567
`depends_on`	→	Taylor's theorem with Lagrange remainder	declared	appendices/A-long-proofs.tex:10567
`depends_on`	←	Uniform quadratic approximation on the $n^{-1/2}$ scale	declared	appendices/A-long-proofs.tex:10627
`depends_on`	←	Root-$n$ localisation	declared	appendices/A-long-proofs.tex:10680
`proves`	←	app:A-long-proofs@proof-127	declared	appendices/A-long-proofs.tex:10570