Estimating the Dimension of a Model
The Annals of Statistics, 1978 · paper · Mathematics and statistics
Treats the choice among models of different dimensions as a Bayes problem and derives the leading terms of its large-sample expansion. The resulting rule picks the model maximizing the log likelihood minus half the dimension times log n, independent of the prior.
Checked: PDF from sites.stat.washington.edu, fetched 5 Oct 2026. Page numbers are those of that PDF.
2 findings
Reference or label Reference [1] prints the title as "A new look at the statistical identification model"; the cited paper's title is understood to be "A new look at the statistical model identification".
- p. 5 References, entry [1]
A new look at the statistical identification model
- p. 5 Reference [1]
A new look at the statistical identification model.
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Akaike's 1974 paper in IEEE Trans. Automatic Control 19, 716-723 is titled "A new look at the statistical model identification". The page 5 image prints "A new look at the statistical identification model. IEEE Trans. Auto. Control 19 716-723", so the quoted wording is in the print and is not an extraction reordering. The true title cannot be checked from the files provided.
Second check: another model agreed
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Wording The proof of Lemma 3 says "Therefore Y ∘ θ − b(θ) is strictly convex, and is easily seen to attain its maximum" and later "By strict convexity it is now easy to determine ρ"; the function is strictly concave.
- p. 4 Proof of Lemma 3
Therefore Y ∘ θ − b(θ) is strictly
- p. 4 Proof of Lemma 3, last sentence
By strict convexity it is now easy to determine
- p. 4 Proof of Lemma 3
Therefore Y o 6 - b(6) is strictly
Only the authors can say what is right here.
Judge's note: Read on the page 4 image. The preceding sentence states that the matrix of second-order derivatives of b(θ) is the covariance matrix of y and is positive definite. Reworking: Y ∘ θ is linear in θ, so the Hessian of Y ∘ θ − b(θ) is minus the Hessian of b(θ), which is negative definite. The function is therefore strictly concave. This is also what the rest of the argument uses: a maximum A, and bounds of the form A − λ||θ − θ0||² with λ positive. The word "convex" is the opposite of what the adjacent lines establish. Lemma 3 as stated is not affected. The paper gives no qualifier such as "convex upward" that would make the word correct.
Second check: another model agreed
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6 minor (typos and the like; most checked by the judge only)
Wording Lemma 3 states "A − λ1||θ − θ0||² < (Y ∘ θ − b(θ)) < A − λ2||θ − θ0||²" with strict inequalities "wherever exp(Y ∘ θ − b(θ)) > ρ"; at θ = θ0, which lies in that region, all three expressions equal A.
- p. 4 Lemma 3, displayed inequality
< (Y ∘ θ − b(θ)) < A − λ2||θ − θ0||²
- p. 4 Proof of Lemma 3
be the point where the
Only the authors can say what is right here.
Judge's note: Read on the page 4 image. Both signs in the display are a plain "<"; the same page prints "≦" with a bar in the proof of Lemma 2, so the bar has not been lost in the scan. The proof takes θ0 to be the point where the maximum A is attained. Since ρ < e^A, exp(Y ∘ θ0 − b(θ0)) = e^A > ρ, so θ0 is in the stated region. At θ0 the left side is A − λ1·0 = A, the middle is A, and the right side is A − λ2·0 = A, so the display reads A < A < A, which is false. The inequalities hold if they are made non-strict or if θ0 is excluded, and the use made of the lemma in the Proposition is unchanged.
Second check: another model agreed
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Typo Section 1 prints "by studying the asymptotic beavior of Bayes estimators"; "beavior" should be "behavior".
- p. 2 Section 1, fourth paragraph
asymptotic beavior of Bayes estimators
- p. 2 Section 1 Introduction, 4th paragraph
by studying the asymptotic beavior of Bayes estimators
Judge's note: Read on the page 2 image, fourth paragraph of the Introduction: the printed word is "beavior". Not an extraction artefact.
Second check: none (judge only)
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Typo Section 1 prints "put positive probability on some lower-dimentional subspaces of the parameter space"; "dimentional" should be "dimensional".
- p. 2 Section 1, fourth paragraph
lower-dimentional subspaces
- p. 2 Section 1 Introduction, 4th paragraph
put positive probability on some lower-dimentional subspaces
- p. 4 closing discussion
towards lower-dimensional
- p. 2
lower-dimentional subspaces of the parameter
Judge's note: Read on the page 2 image: the printed word is "lower-dimentional". The page 4 image prints "lower-dimensional models" with the usual spelling, so the page 2 form is a misspelling and not a house style.
Second check: none (judge only)
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Typo Section 2 prints "a k_j-dimensional density that is bounded and locally bounded away form zero throughout m_j ∩ Θ"; "form" should be "from".
- p. 3 Section 2, second paragraph
locally bounded away form zero
- p. 3 Section 2, 2nd paragraph
bounded and locally bounded away form zero
- p. 3
locally bounded away form zero throughout
Judge's note: Read on the page 3 image, second paragraph of Section 2: the printed word is "form". The context ("bounded away ... zero") requires "from".
Second check: none (judge only)
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Typo Section 3 opens "The asymptotic expansion of S(y, n, j)" with a lower-case bold y, while the function is defined as "S(Y, n, j)" with Y "the averaged y-statistic (1/n) Σ y(X_i)" and is written S(Y, n, j) in the Proposition.
- p. 3 Section 3, first sentence
The asymptotic expansion of S(y, n, j)
- p. 3 Section 2, displayed definition
S(Y, n, j) = log
- p. 3 Section 2, density f(x,θ)
exp(θ · y(x) − b(θ))
- p. 3 Section 2, definition of S
S(Y, n, j) = log ∫ αj exp((Y ∘ θ − b(θ))n) dμj(θ)
Judge's note: On the page 3 image the argument in the first sentence of Section 3 is a bold lower-case y. The definition a few lines above and the Proposition directly below both print a bold capital Y. The paper defines y as the sufficient statistic of one observation and Y as its average, so the lower-case letter names a different object. The meaning is clear from context, so this is a notation slip. The dot-versus-circle point also raised in this cluster is judged under C006.
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Typo The inner product is written with a centred dot in the density, "f(x, θ) = exp(θ · y(x) − b(θ))", and with a small circle from the definition of S onward, "exp((Y ∘ θ − b(θ))n)"; the circle is never defined.
- p. 3 Section 2, displayed density
f(x, θ) = exp(θ · y(x) − b(θ))
- p. 3 Section 2, displayed definition of S
exp((Y ∘ θ − b(θ))n)
Judge's note: Both forms are on the page 3 image: a centred dot in the displayed density and a small circle in the definition of S, the Proposition and Lemma 1. The page 4 image uses the circle throughout Lemma 3. Both plainly mean the Euclidean inner product, so nothing is misstated. It is an undefined change of symbol and either form could stand.
Second check: none (judge only)