{
 "claims": [
  {
   "claim": "Under consistency, unconfoundedness and positivity with a continuous covariate, and without further assumptions, there is no uniformly consistent estimator of $\\mu=\\mathrm{E}[Y(1)]$, and hence no uniformly consistent estimator of the average treatment effect.",
   "claim_id": "P046-H1",
   "related_corrections": [
    "P046-E004"
   ],
   "tied_to": "Proposition 1 (attributed to Theorem 3 of Robins and Ritov 1997; no proof in the paper); abstract"
  },
  {
   "claim": "When the propensity score is known, the Horvitz–Thompson estimator is unbiased, uniformly root-$n$ consistent and asymptotically normal for $\\mu$, regardless of the complexity of the propensity-score function.",
   "claim_id": "P046-H2",
   "related_corrections": [],
   "tied_to": "Proposition 2; Figure 3"
  },
  {
   "claim": "With a binary outcome and a known propensity score, a finite-sample-valid confidence interval for $\\mu$ has width of order $n^{-1/2}$ regardless of the complexity of the score. The paper gives $\\hat\\mu\\pm\\sqrt{\\xi\\log(2/\\alpha)/(2n)}$, where $\\xi=n^{-1}\\sum_i\\pi(W_i)^{-2}$.",
   "claim_id": "P046-H3",
   "related_corrections": [
    "P046-E001"
   ],
   "tied_to": "Proposition 3 and its proof in Appendix A; abstract"
  },
  {
   "claim": "With a known propensity score, the cross-fitted regression-adjusted estimator $\\hat\\mu_R$ is finite-sample unbiased. If the fitted regressions converge in mean square, it is root-$n$ consistent even when their limits are misspecified. It is asymptotically at least as efficient as Horvitz–Thompson when a finite-parameter outcome model is fitted by weighted least squares with weights $(1-\\pi)/\\pi^2$, and semiparametrically efficient when the outcome regression is estimated consistently.",
   "claim_id": "P046-H4",
   "related_corrections": [
    "P046-E002",
    "P046-E003"
   ],
   "tied_to": "Proposition 4(a)-(d); proof of 4(a) in Appendix A."
  },
  {
   "claim": "In the reported simulations, the RMSE of estimators that do not use the true propensity score does not decrease with $n$ under DGP 3, and the logistic-based estimators are inconsistent under DGP 2. HT and HTLOESS, which use the true score, converge at the parametric rate under all three designs.",
   "claim_id": "P046-H5",
   "related_corrections": [
    "P046-E005"
   ],
   "tied_to": "Figure 3; Section 3.3."
  },
  {
   "claim": "Regression adjustment with the true propensity score helps when the outcome regression can be learned (HTLOESS beats HT in DGP 1 and 2) and hurts when it cannot (HT beats HTLOESS in DGP 3), while the RMSE of HT is essentially the same in all three DGPs.",
   "claim_id": "P046-H6",
   "related_corrections": [],
   "tied_to": "Figure 3; Section 3.3"
  },
  {
   "claim": "Randomized controlled trials are special among studies with unconfounded treatment: a successfully executed trial has a statistical advantage beyond enforcing unconfoundedness, because it supplies the propensity score, so estimation and inference are fundamentally harder in observational settings.",
   "claim_id": "P046-H7",
   "related_corrections": [
    "P046-E001",
    "P046-E002",
    "P046-E004"
   ],
   "tied_to": "Propositions 1-4 and Figure 3; abstract; Section 4"
  },
  {
   "claim": "Recording the assignment probabilities is enough to obtain the known-propensity benefits in natural or automated experiments; the analyst need not recover the complete assignment function.",
   "claim_id": "P046-H8",
   "related_corrections": [
    "P046-E001"
   ],
   "tied_to": "Propositions 2-4; Section 4 (Discussion)."
  }
 ],
 "date_published": "2026-10-25",
 "findings": [
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of the heading Abstract and the first eight lines of the abstract. The third sentence reads: First, in settings with at least one continuous confounder, there exists no estimator of the average treatment effect that is uniformly consistent unless the propensity score is known or additional assumptions are made on the complexity of the propensity score function. A red outline surrounds the words the propensity score function, at the start of the sixth line.",
     "caption": "Abstract, p. 3 (PDF p. 2). The outline marks the phrase that names the propensity score function as the only object of the additional assumptions.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E004.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P046-T10",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T10"
    }
   ],
   "headline": "Assumptions on the outcome regression alone also permit a uniformly consistent estimator; the abstract names only assumptions on the propensity score function.",
   "id": "P046-E004",
   "location": "Abstract, third sentence, p. 3 (PDF p. 2)",
   "pdf_page": 2,
   "printed_page": "3",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [
    "P046-T10"
   ],
   "text": {
    "as_printed": "First, in settings with at least one continuous confounder, there exists no estimator of the average treatment effect that is uniformly consistent unless the propensity score is known or additional assumptions are made on the complexity of \\hl{the propensity score function}.",
    "corrected": "First, in settings with at least one continuous confounder, there exists no estimator of the average treatment effect that is uniformly consistent unless the propensity score is known or additional assumptions are made on the complexity of \\hl{the outcome regression and/or propensity score functions}.",
    "intended_meaning": "The sentence summarizes Proposition 1, which states the impossibility ``Without further assumptions'', and with this meaning it is correct. Read literally, the clause after ``unless'' lists every remedy, and the list is incomplete: suitable assumptions on the outcome regression alone also suffice, with the propensity score unknown and unrestricted. For example, suppose that $\\mathrm{E}[Y_i(x) \\mid W_i = w] = a_x + h(w)$ for $x \\in \\{0,1\\}$, where $h(w) = \\min\\{1, \\max\\{0, w\\}\\}/4$ is known and $a_0, a_1 \\in [1/4, 1/2]$ are unknown. Then the difference of the within-arm means of $Y_i - h(W_i)$ (zero for an empty arm) estimates the average treatment effect with mean squared error at most $(3/4)^n + 25/[2(n+1)]$, for every measurable propensity score such that $1/4 < \\pi(W_i) < 3/4$. The body of the paper allows for such assumptions (the paragraph after Proposition 1 and footnote 1, p.~6), and no result depends on the literal reading."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-E004",
   "work": {
    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
   },
   "work_id": "P046"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "page_check",
   "exhibits": [
    {
     "alt": "Crop of three lines. The passage begins: trade-off: when the assignment mechanism is unknown, the strength of our statistical procedures critically depend on the assumptions brought to bear. A red outline surrounds the word depend.",
     "caption": "Item 1: Section 2, p. 6 (PDF p. 5). The outline marks “depend”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007.png"
    },
    {
     "alt": "Crop of three lines. The passage includes: To ground these ideas, we consider a regression-based estimator of mu that use a simple type of cross-fitting. A red outline surrounds the word use.",
     "caption": "Item 2: Section 2, p. 7 (PDF p. 6). The outline marks “use”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-2.png"
    },
    {
     "alt": "Crop of three lines. The passage includes: in Proposition 4, where g-hat-0 and g-hat-1 are cross-fitted LOESS regression of the same type as above. A red outline surrounds the word regression.",
     "caption": "Item 3: Section 3.2, p. 10 (PDF p. 9). The outline marks “regression”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-3.png"
    },
    {
     "alt": "Crop of three lines. The sentence reads: For the third data generating process, which is the highly discontinuous function, only the two estimators that use of the true propensity score perform well. A red outline surrounds the words use of.",
     "caption": "Item 4: Section 3.3, p. 11 (PDF p. 10). The outline marks “use of”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-4.png"
    },
    {
     "alt": "Crop of three lines. The sentence reads: The explanation is that it is essentially impossible to capture the outcome regression function here, and trying to do so introduce unnecessary noise, which makes the precision worse. A red outline surrounds the word introduce.",
     "caption": "Item 5: Section 3.3, p. 11 (PDF p. 10). The outline marks “introduce”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-5.png"
    },
    {
     "alt": "Crop of three lines. The sentence reads: Knowledge of the propensity score can enormously beneficial in such settings. A red outline surrounds the words can enormously beneficial, at the start of the second line.",
     "caption": "Item 6: Section 4, p. 12 (PDF p. 11). The outline marks “can enormously beneficial”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-6.png"
    },
    {
     "alt": "Crop of three lines. The sentence reads: The estimation difficulties are often profound because the input space tend to be high-dimensional, and the treatment assignment function and the outcome regression tend to be highly non-smooth with discontinuities. A red outline surrounds the first word tend, at the start of the second line.",
     "caption": "Item 7: Section 4, p. 12 (PDF p. 11). The outline marks “tend”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-7.png"
    },
    {
     "alt": "Crop of the three lines of the reference by P. M. Aronow and Joel A. Middleton, A class of unbiased estimators of the average treatment effect in randomized experiments, Journal of Causal Inference, 2013. The third line reads: 2013. doi: doi:10.1515/jci-2012-0009, followed by the URL. A red outline surrounds the words doi: doi:.",
     "caption": "Item 8: References, p. 13 (PDF p. 12). The outline marks the repeated label “doi: doi:”.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E007-8.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [],
   "headline": "Seven sentences have errors of agreement or wording, and one reference repeats the label “doi:”.",
   "id": "P046-E007",
   "location": "Seven sentences on pp. 6, 7, 10, 11 and 12 (PDF pp. 5, 6, 9, 10 and 11), and the Aronow–Middleton entry of the reference list, p. 13 (PDF p. 12)",
   "pdf_page": 5,
   "printed_page": "6",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [],
   "text": {
    "as_printed": "\\begin{enumerate}\n\\item \\emph{p.~6.} the strength of our statistical procedures critically \\hl{depend} on the assumptions brought to bear.\n\\item \\emph{p.~7.} we consider a regression-based estimator of $\\mu$ that \\hl{use} a simple type of cross-fitting.\n\\item \\emph{p.~10.} where $\\hat g_0$ and $\\hat g_1$ are cross-fitted LOESS \\hl{regression} of the same type as above.\n\\item \\emph{p.~11.} only the two estimators that \\hl{use of} the true propensity score perform well.\n\\item \\emph{p.~11.} and trying to do so \\hl{introduce} unnecessary noise\n\\item \\emph{p.~12.} Knowledge of the propensity score \\hl{can enormously} beneficial in such settings.\n\\item \\emph{p.~12.} because the input space \\hl{tend} to be high-dimensional\n\\item \\emph{p.~13.} Peter M. Aronow and Joel A. Middleton. A class of unbiased estimators of the average treatment effect in randomized experiments. \\emph{Journal of Causal Inference}, 1(1):135--154, 2013. \\hl{doi: doi:}10.1515/jci-2012-0009. URL \\url{https://doi.org/10.1515/jci-2012-0009}.\n\\end{enumerate}",
    "corrected": "\\begin{enumerate}\n\\item the strength of our statistical procedures critically \\hl{depends} on the assumptions brought to bear.\n\\item we consider a regression-based estimator of $\\mu$ that \\hl{uses} a simple type of cross-fitting.\n\\item where $\\hat g_0$ and $\\hat g_1$ are cross-fitted LOESS \\hl{regressions} of the same type as above.\n\\item only the two estimators that \\hl{use} the true propensity score perform well.\n\\item and trying to do so \\hl{introduces} unnecessary noise\n\\item Knowledge of the propensity score \\hl{can be enormously} beneficial in such settings.\n\\item because the input space \\hl{tends} to be high-dimensional\n\\item P. M. Aronow and Joel A. Middleton. A class of unbiased estimators of the average treatment effect in randomized experiments. \\emph{Journal of Causal Inference}, 1(1):135--154, 2013. \\hl{doi:} 10.1515/jci-2012-0009. URL \\url{https://doi.org/10.1515/jci-2012-0009}.\n\\end{enumerate}",
    "evidence": "I checked the pages. On p.~7 the paper calls $\\hat g_0$ and $\\hat g_1$ ``two estimators of the regression function'', which supports the plural in item 3.",
    "problem": "Items 1 to 7 are errors of grammar: a singular subject with a plural verb in items 1, 2, 5 and 7, a singular noun for the two fitted regressions in item 3, an extra ``of'' in item 4, and a missing ``be'' in item 6. The reference in item 8 prints the label ``doi:'' twice.",
    "what_survives": "The intended meaning is clear in every case, and no statement, formula or number changes."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-E007",
   "work": {
    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
   },
   "work_id": "P046"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of five lines. The first two lines end the preceding paragraph: This confidence interval is finite-sample valid, meaning that its coverage is guaranteed to be greater or equal to its nominal level even in small samples. Proposition 3 follows: Suppose the propensity scores are known. Then a finite-sample-valid 1 minus alpha confidence interval for mu is mu-hat plus or minus the square root of xi log(2/alpha)/2n, where xi is the mean of the square inverse of the propensity scores. A red outline surrounds the square root of xi log(2/alpha)/2n.",
     "caption": "Proposition 3, p. 7 (PDF p. 6), with the two lines that precede it. The outline marks the radius of the printed interval.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E001.png"
    },
    {
     "alt": "Crop of three lines of text and one displayed inequality. The text reads: The Horvitz-Thompson estimator can be written mu-hat equals the sum of the A-i, where A-i equals X-i Y-i divided by n pi(W-i). Because the outcome is binary, the bound 0 at most A-i at most 1 over n pi(W-i) holds with probability one. Recall that E[mu-hat] = mu, so by Hoeffding's inequality. The display states that the probability, conditional on W, that the absolute value of mu-hat minus mu is at least the square root of xi log(2/alpha)/2n is at most 2 exp of minus 2 xi log(2/alpha)/2n divided by the sum over i of n pi(W-i) to the power minus 2. Red outlines surround E[mu-hat] = mu in the text and the second mu inside the absolute value in the display.",
     "caption": "Proof of Proposition 3, Appendix A, p. 16 (PDF p. 15). The outlines mark the unconditional identity $\\mathrm{E}[\\hat\\mu] = \\mu$ that the proof recalls and the centre $\\mu$ of the conditional bound.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E001-proof.png"
    },
    {
     "alt": "Two panels that share a horizontal axis, K from 0 to 100, under the title: Proposition 3 as printed: the probability that mu is not in the interval equals 1.198 times 10 to the minus 7 at nominal level alpha equal to 10 to the minus 7. The top panel is a blue curve, on a logarithmic vertical axis, of the probability that mu is not in the interval given K. The curve equals 1 for K below about 17, falls to about 3 times 10 to the minus 13 near K equal to 64, and rises again to about 0.3 at K equal to 100. A dashed orange horizontal line marks the nominal level 10 to the minus 7; the curve is above this line for K at most 50 and for K at least 72. A dotted vertical line at K equal to 60 is labelled: the conditional mean of mu-hat equals mu. The bottom panel is a bar chart of the probability that K equals k and mu is not in the interval, in units of 10 to the minus 9. The bars form two groups, one between K equal to 29 and 54 with a maximum of about 12.9 at K equal to 44, and a smaller one between K equal to 70 and 86 with a maximum of about 5 at K equal to 76. A label gives the exact sum over K, 1.198 times 10 to the minus 7, above the nominal level, 1.000 times 10 to the minus 7.",
     "caption": "The counterexample for P046-E001 ($n = 100$, $\\alpha = 10^{-7}$), by the number $K$ of units such that $W_i \\le 3/5$. Top: the probability, given $K$, that the printed interval fails to contain $\\mu$, on a logarithmic scale; the dashed line is the nominal level $\\alpha$. Bottom: the contribution of each value of $K$ to that probability; the contributions sum to $1.198 \\times 10^{-7}$. Given $K$, the estimator $\\hat\\mu$ has mean $K/100$, which equals $\\mu = 3/5$ only when $K = 60$.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E001-noncoverage.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P046-T1",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T1"
    },
    {
     "id": "P046-T2",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T2"
    },
    {
     "id": "P046-T3",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T3"
    },
    {
     "id": "P046-T6",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T6"
    }
   ],
   "headline": "The interval of Proposition 3 is not finite-sample valid at every level, and its proof centres a conditional Hoeffding bound at $\\mu$ where the conditional mean of $\\hat\\mu$ is required; the interval with radius $(\\sqrt{\\xi}+1)\\sqrt{\\log(4/\\alpha)/(2n)}$ is valid.",
   "id": "P046-E001",
   "location": "Proposition 3, p. 7 (PDF p. 6); its proof in Appendix A, pp. 15–16 (PDF pp. 14–15), at the display after “Recall that $\\mathrm{E}[\\hat\\mu] = \\mu$, so by Hoeffding’s inequality,” on p. 16",
   "pdf_page": 6,
   "printed_page": "7",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Moderate",
   "subject": "paper",
   "target_ids": [
    "P046-T1",
    "P046-T2",
    "P046-T3",
    "P046-T6"
   ],
   "text": {
    "as_printed": "\\emph{Proposition 3 (p.~7).} Suppose the propensity scores are known. Then a finite-sample-valid $1-\\alpha$ confidence interval for $\\mu$ is $\\hat\\mu \\pm \\hl{\\sqrt{\\xi \\log(2/\\alpha)/2n}}$, where $\\xi = n^{-1}\\sum_{i=1}^n [\\pi(W_i)]^{-2}$ is the mean of the square inverse of the propensity scores.\n\n\\emph{Proof, Appendix A (p.~16).} The Horvitz--Thompson estimator can be written $\\hat\\mu = \\sum_{i=1}^n A_i$, where $A_i = [n\\pi(W_i)]^{-1}X_iY_i$. Because the outcome is binary, the bound $0 \\le A_i \\le [n\\pi(W_i)]^{-1}$ holds with probability one. Recall that $\\mathrm{E}[\\hat\\mu] = \\mu$, \\hl{so by Hoeffding's inequality},\n\\[\n\\Pr\\Bigl( |\\hat\\mu - \\hl{\\mu}| \\ge \\sqrt{\\xi \\log(2/\\alpha)/2n} \\Bigm| \\mathbf{W} \\Bigr) \\le 2\\exp\\biggl( \\frac{-2\\xi \\log(2/\\alpha)/2n}{\\sum_{i=1}^n [n\\pi(W_i)]^{-2}} \\biggr).\n\\]",
    "corrected": "\\emph{Proposition 3.} Suppose the propensity scores are known, and let $0 < \\alpha < 1$. Under the assumptions of Section 2 (an i.i.d.\\ sample, binary $Y_i$ and $X_i$, consistency, unconfoundedness and positivity), a finite-sample-valid $1-\\alpha$ confidence interval for $\\mu = \\mathrm{E}[Y_i(1)]$ is\n\\[\n\\hat\\mu \\pm \\hl{(\\sqrt{\\xi}+1)\\sqrt{\\log(4/\\alpha)/(2n)}},\n\\]\nwhere $\\xi = n^{-1}\\sum_{i=1}^n [\\pi(W_i)]^{-2}$ is the mean of the square inverse of the propensity scores. \\hl{If a number $b > 0$ is known such that $\\pi(W_i) \\ge b$ with probability one, then $\\hat\\mu \\pm b^{-1}\\sqrt{\\log(2/\\alpha)/(2n)}$ is also a finite-sample-valid $1-\\alpha$ confidence interval for $\\mu$.}\n\n\\emph{Proof, in place of the conditional display.} Let $m(w) = \\mathrm{E}[Y_i(1) \\mid W_i = w]$ be the outcome regression function. Recall that\n\\[\n\\hl{\\mathrm{E}[\\hat\\mu \\mid \\mathbf{W}] = \\mu_{\\mathbf{W}}}, \\qquad \\text{where } \\mu_{\\mathbf{W}} = \\frac{1}{n}\\sum_{i=1}^n m(W_i),\n\\]\nso by Hoeffding's inequality, with $c = \\sqrt{\\log(4/\\alpha)/(2n)}$,\n\\[\n\\Pr\\Bigl( |\\hat\\mu - \\hl{\\mu_{\\mathbf{W}}}| \\ge \\sqrt{\\xi}\\,c \\Bigm| \\mathbf{W} \\Bigr) \\le 2\\exp\\biggl( \\frac{-2\\xi c^2}{\\sum_{i=1}^n [n\\pi(W_i)]^{-2}} \\biggr) = \\frac{\\alpha}{2}.\n\\]\n\\hl{The variables $m(W_i)$ are i.i.d., take values in $[0,1]$ and have mean $\\mu$, so a second application of Hoeffding's inequality gives $\\Pr(|\\mu_{\\mathbf{W}} - \\mu| \\ge c) \\le \\alpha/2$. The triangle inequality and a union bound then give $\\Pr(|\\hat\\mu - \\mu| \\ge (\\sqrt{\\xi}+1)c) \\le \\alpha$.}",
    "evidence": "The evidence is an exact counterexample to the printed statement and a derivation of the corrected one.\n\n\\emph{Counterexample.} Let $n = 100$ and $\\alpha = 10^{-7}$. Let $W_i$ be uniform on $(0,1)$, and set $Y_i(1) = \\mathbf{1}\\{W_i \\le 3/5\\}$ and $Y_i(0) = 0$. Let $\\pi(w) = 13/20$ for $w \\le 3/5$ and $\\pi(w) = 99/100$ for $w > 3/5$, and assign the treatments independently given the covariates. The observations are i.i.d., the outcome and the treatment are binary, the covariate is continuous, consistency and unconfoundedness hold, positivity holds with $\\delta = 1/200$, and $\\mu = 3/5$. Let $K$ be the number of units such that $W_i \\le 3/5$, and let $S$ be the number of these units that are treated. In this design $\\hat\\mu = S/65$ and $\\xi = [400K/169 + 10000(100-K)/9801]/100$, which implies that the printed interval depends on the sample only through $(K, S)$. For $0 \\le s \\le k \\le 100$, I have\n\\[\n\\Pr(K = k,\\, S = s) = \\binom{100}{k}\\binom{k}{s}\\frac{39^s\\, 21^{k-s}\\, 40^{100-k}}{100^{100}}.\n\\]\nThe printed interval fails to contain $\\mu$ exactly when $200(S/65 - 3/5)^2/\\xi > \\log(2 \\cdot 10^7)$. This inequality holds for 1946 of the 5151 pairs $(k,s)$. Summing their probabilities in exact rational arithmetic, with rational bounds on the logarithm, I obtain\n\\[\n\\Pr\\Bigl( |\\hat\\mu - \\mu| > \\sqrt{\\xi \\log(2/\\alpha)/2n} \\Bigr) = 1.19792910697956\\ldots \\times 10^{-7} > 10^{-7} = \\alpha.\n\\]\nThe conditional display in the proof fails more directly. On the event that $W_i > 3/5$ for every $i$, which has probability $(2/5)^{100} > 0$, the estimator $\\hat\\mu$ equals $0$ and $\\xi = (100/99)^2$. On this event $|\\hat\\mu - \\mu| = 3/5$ exceeds the printed radius, the left side of the display equals $1$, and the right side equals $10^{-7}$.\n\nThe table lists, at three levels, the exact probability that each interval fails to contain $\\mu$ in this design. Only the first entry of the first row exceeds its nominal level.\n\\begin{audittable}{lll}{Exact probability that the interval fails to contain $\\mu$ in the counterexample ($n = 100$), for the printed radius and for the corrected radius.}\nNominal level $\\alpha$ & Printed interval & Corrected interval \\\\\n$10^{-7}$ & $\\hl{1.198 \\times 10^{-7}}$ & $1.207 \\times 10^{-23}$ \\\\\n$0.05$ & $0.01381$ & $2.324 \\times 10^{-6}$ \\\\\n$0.01$ & $0.003614$ & $2.396 \\times 10^{-8}$ \\\\\n\\end{audittable}\n\n\\emph{Derivation of the corrected interval.} Fix $\\alpha \\in (0,1)$ and set $c = \\sqrt{\\log(4/\\alpha)/(2n)}$. By consistency, I have $X_iY_i = X_iY_i(1)$. Using unconfoundedness and the independence of the units, I find\n\\[\n\\mathrm{E}[A_i \\mid \\mathbf{W}] = \\frac{\\mathrm{E}[X_i \\mid W_i]\\,\\mathrm{E}[Y_i(1) \\mid W_i]}{n\\pi(W_i)} = \\frac{m(W_i)}{n},\n\\]\nand summing over $i$, I obtain $\\mathrm{E}[\\hat\\mu \\mid \\mathbf{W}] = \\mu_{\\mathbf{W}}$. Conditional on $\\mathbf{W}$, the variables $A_i$ are independent and satisfy $0 \\le A_i \\le [n\\pi(W_i)]^{-1}$. Since $\\sum_{i=1}^n [n\\pi(W_i)]^{-2} = \\xi/n$, Hoeffding's inequality implies that\n\\[\n\\Pr\\bigl( |\\hat\\mu - \\mu_{\\mathbf{W}}| \\ge t \\bigm| \\mathbf{W} \\bigr) \\le 2\\exp(-2nt^2/\\xi)\n\\]\nfor every $t > 0$ that is a function of $\\mathbf{W}$. Setting $t = \\sqrt{\\xi}\\,c$ and taking the expectation over $\\mathbf{W}$, I obtain\n\\[\n\\Pr\\bigl( |\\hat\\mu - \\mu_{\\mathbf{W}}| \\ge \\sqrt{\\xi}\\,c \\bigr) \\le 2\\exp(-2nc^2) = \\alpha/2.\n\\]\nNext, the variables $m(W_i)$ are i.i.d., take values in $[0,1]$ and have mean $\\mu$. Applying Hoeffding's inequality to their mean, I have $\\Pr(|\\mu_{\\mathbf{W}} - \\mu| \\ge c) \\le 2\\exp(-2nc^2) = \\alpha/2$. If neither event occurs, then by the triangle inequality I have\n\\[\n|\\hat\\mu - \\mu| \\le |\\hat\\mu - \\mu_{\\mathbf{W}}| + |\\mu_{\\mathbf{W}} - \\mu| < (\\sqrt{\\xi}+1)c.\n\\]\nBy a union bound, I conclude that $\\Pr(|\\hat\\mu - \\mu| \\ge (\\sqrt{\\xi}+1)c) \\le \\alpha$. Positivity implies that $\\sqrt{\\xi} < \\delta^{-1}$, and the width of the corrected interval is $O(n^{-1/2})$ for fixed $\\alpha$. For the alternative with a known bound $b > 0$ such that $\\pi(W_i) \\ge b$ with probability one, the variables $X_iY_i/\\pi(W_i)$ are i.i.d., take values in $[0, b^{-1}]$ and have mean $\\mu$, and Hoeffding's inequality applied to their mean $\\hat\\mu$ gives the radius $b^{-1}\\sqrt{\\log(2/\\alpha)/(2n)}$.",
    "problem": "The proof conditions on the covariate vector $\\mathbf{W} = (W_1, \\dots, W_n)$ and applies Hoeffding's inequality to the deviation of $\\hat\\mu$ from $\\mu$. Conditional on $\\mathbf{W}$, however, the mean of $\\hat\\mu$ need not equal $\\mu$. Let $m(w) = \\mathrm{E}[Y_i(1) \\mid W_i = w]$ and $\\mu_{\\mathbf{W}} = n^{-1}\\sum_{i=1}^n m(W_i)$. Then $\\mathrm{E}[\\hat\\mu \\mid \\mathbf{W}] = \\mu_{\\mathbf{W}}$; taking the expectation over $\\mathbf{W}$ gives the unconditional identity $\\mathrm{E}[\\hat\\mu] = \\mu$ that the proof recalls. Hoeffding's inequality bounds the conditional probability that $|\\hat\\mu - \\mu_{\\mathbf{W}}|$ exceeds the printed radius. It does not bound $|\\hat\\mu - \\mu|$, and the printed radius does not account for the variation of $\\mu_{\\mathbf{W}}$ around $\\mu$.\n\nThe proposition itself fails as stated. It claims coverage at least $1-\\alpha$ and states no restriction on $\\alpha$. In the design that I give below, which satisfies every assumption of Section 2, the printed interval fails to contain $\\mu$ with probability $1.198 \\times 10^{-7}$ at $\\alpha = 10^{-7}$. This level is far from the levels used in practice. The counterexample does not show that the printed interval undercovers at the 95\\% or 99\\% level: in the same design the probability that it fails to contain $\\mu$ is below $\\alpha$ at both levels. These calculations do not settle whether the printed interval is valid at such levels in every design. To repair the proposition I keep the estimand, the estimator and the assumptions, and I widen the interval. A second Hoeffding bound controls $\\mu_{\\mathbf{W}} - \\mu$, and I allot error probability $\\alpha/2$ to each of the two bounds. The corrected interval still has width of order $n^{-1/2}$.",
    "what_survives": "The claim of the abstract and of p.~7 survives: with a binary outcome and a known propensity score, there exist finite-sample valid confidence intervals for $\\mu$ whose width decreases at the rate $n^{-1/2}$. The corrected interval is one. The printed radius remains valid when the propensity score is constant, by unconditional Hoeffding's inequality. Propositions 1, 2 and 4 and the numerical illustration of Section 3 do not use Proposition 3. What changes is the radius in Proposition 3, which becomes larger, the statement on p.~7 that the printed interval guarantees uniform validity, and the proof in Appendix A."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-E001",
   "work": {
    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
   },
   "work_id": "P046"
  },
  {
   "before_review": null,
   "class": "error",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of Proposition 4 from its first line through part (c). The first line reads: Proposition 4. Suppose the propensity scores are known. Then. Part (a): mu-hat-R is finite-sample unbiased for mu. Part (b): If g-hat-0 and g-hat-1 are convergent in mean square (but not necessarily consistent for the true outcome regression function), then mu-hat-R is root-n consistent for mu. Part (c): If g-hat-0 and g-hat-1 are parameterized by a finite number of parameters and estimated by minimizing the empirical weighted mean square error of the outcome regression with weights w-i equal to (1 minus pi(W-i)) divided by pi(W-i) squared, then mu-hat-R is at least as asymptotically efficient as the Horvitz-Thompson estimator mu-hat. A red outline surrounds the words parameterized by a finite number of parameters.",
     "caption": "Proposition 4, parts (a) to (c), p. 8 (PDF p. 7). The outline marks the hypothesis of part (c) on the working family.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E002.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P046-T4",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T4"
    },
    {
     "id": "P046-T5",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T5"
    }
   ],
   "headline": "Proposition 4(c) holds when the working family of outcome regressions contains the zero function and the fitted regressions converge; for other families the adjusted estimator can be less efficient than the Horvitz–Thompson estimator.",
   "id": "P046-E002",
   "location": "Proposition 4(c), p. 8 (PDF p. 7)",
   "pdf_page": 7,
   "printed_page": "8",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": "Minor",
   "subject": "paper",
   "target_ids": [
    "P046-T4",
    "P046-T5"
   ],
   "text": {
    "as_printed": "\\textbf{Proposition 4} Suppose the propensity scores are known. Then: \\ldots\n\n(c) If $\\hat g_0$ and $\\hat g_1$ are \\hl{parameterized by a finite number of parameters} and estimated by minimizing the empirical weighted mean square error of the outcome regression with weights $w_i = [\\pi(W_i)]^{-2}[1 - \\pi(W_i)]$, then $\\hat\\mu_R$ is at least as asymptotically efficient as the Horvitz--Thompson estimator $\\hat\\mu$.",
    "corrected": "\\textbf{Proposition 4} Suppose the propensity scores are known. Then: \\ldots\n\n(c) \\hl{Suppose that} $\\hat g_0$ and $\\hat g_1$ are estimated by minimizing the empirical weighted mean square error of the outcome regression with weights $w_i = [\\pi(W_i)]^{-2}[1 - \\pi(W_i)]$ \\hl{over a fixed working family that is parameterized by a finite number of parameters and contains the zero function (or has the zero function in its weighted $L^2$ closure), and that both converge in $L^2$ to a deterministic minimizer over this family of}\n\\[\n\\hl{R(g) = \\mathrm{E}\\biggl[ \\frac{1 - \\pi(W_i)}{\\pi(W_i)}\\bigl( Y_i(1) - g(W_i) \\bigr)^2 \\biggr].}\n\\]\n\\hl{Then} $\\hat\\mu_R$ is at least as asymptotically efficient as the Horvitz--Thompson estimator $\\hat\\mu$.",
    "evidence": "The evidence is an exact counterexample to the printed statement and a derivation of the corrected one.\n\n\\emph{Counterexample.} Let $W_i$ be uniform on $(0,1)$, let $L_i = \\mathbf{1}\\{W_i \\le 1/2\\}$, and let $Z_i$ be a Bernoulli variable with mean $1/10$ that is independent of $W_i$. Set $Y_i(0) = Y_i(1) = L_iZ_i$. Let $X_i$ be independent of $Z_i$ given $W_i$, with $\\pi(W_i) = 1/4$ if $L_i = 1$ and $\\pi(W_i) = 3/4$ if $L_i = 0$. The assumptions of Section 2 hold, with $\\delta = 1/8$ in the positivity condition, and $\\mu = 1/20$. Consider the working family of the functions $g_\\theta(w) = (1/2)\\mathbf{1}\\{w \\le 1/2\\} + \\theta\\,\\mathbf{1}\\{w > 1/2\\}$ with $\\theta \\in \\mathbb{R}$. It has one parameter and does not contain the zero function. The printed weights equal $12$ where $L_i = 1$ and $4/9$ where $L_i = 0$. Since $Y_i = 0$ where $L_i = 0$, the weighted criterion on any training fold equals a term that does not depend on $\\theta$ plus $(4/9)N\\theta^2$, where $N$ is the number of treated units of the training fold such that $L_i = 0$. The value $\\theta = 0$ minimizes it for every training fold (I take $\\theta = 0$ when $N = 0$), and $\\theta = 0$ is the unique minimizer of the population criterion $R(g_\\theta) = 3/8 + \\theta^2/6$. With $g_*(w) = (1/2)\\mathbf{1}\\{w \\le 1/2\\}$, I therefore have $\\hat g_0 = \\hat g_1 = g_*$, and $\\hat\\mu_R$ is the mean of the i.i.d.\\ variables $A_i = X_i[Y_i - g_*(W_i)]/\\pi(W_i) + g_*(W_i)$. The table lists the values of $A_i$ and of the Horvitz--Thompson summand $H_i = X_iY_i/\\pi(W_i)$.\n\\begin{audittable}{llll}{The summands of the two estimators in the counterexample.}\nEvent & Probability & $A_i$ & $H_i$ \\\\\n$L_i = 1$, $X_i = 0$ & $3/8$ & $1/2$ & $0$ \\\\\n$L_i = 1$, $X_i = 1$, $Z_i = 0$ & $9/80$ & $-3/2$ & $0$ \\\\\n$L_i = 1$, $X_i = 1$, $Z_i = 1$ & $1/80$ & $5/2$ & $4$ \\\\\n$L_i = 0$ & $1/2$ & $0$ & $0$ \\\\\n\\end{audittable}\nA direct computation gives $\\mathrm{E}[A_i] = \\mathrm{E}[H_i] = 1/20$, $\\mathrm{E}[A_i^2] = 17/40$ and $\\mathrm{E}[H_i^2] = 1/5$. For every $n$, I obtain\n\\[\n\\operatorname{Var}(\\hat\\mu_R) = \\frac{169}{400n} > \\frac{79}{400n} = \\operatorname{Var}(\\hat\\mu).\n\\]\nIn this example $\\operatorname{Var}(Y_i(1)) = 19/400$, $R(0) = 3/20$ and $R(g_*) = 3/8$.\n\n\\emph{Derivation of the corrected statement.} Fix a square-integrable function $g$ and write $A_g = Y_i(1) + (X_i/\\pi(W_i) - 1)(Y_i(1) - g(W_i))$ for the summand of $\\hat\\mu_R$. By unconfoundedness, conditional on $(W_i, Y_i(1))$ the variable $A_g$ has mean $Y_i(1)$ and variance $(1 - \\pi(W_i))\\pi(W_i)^{-1}(Y_i(1) - g(W_i))^2$. By the law of total variance, I have $\\operatorname{Var}(A_g) = \\operatorname{Var}(Y_i(1)) + R(g)$. For $g = 0$ this is the variance of the Horvitz--Thompson summand. If the working family contains the zero function, then a minimizer $g_*$ of $R$ over the family satisfies $R(g_*) \\le R(0)$. Suppose that $\\hat g_0$ and $\\hat g_1$ converge to $g_*$ in $L^2$. Conditional on the training fold, the summands of the difference between $\\hat\\mu_R$ and the mean of the variables $A_{g_*}$ have mean zero and second moment $\\mathrm{E}[(1 - \\pi(W_i))\\pi(W_i)^{-1}(\\hat g_j(W_i) - g_*(W_i))^2]$ over the evaluation fold, which implies that this difference is $o_p(n^{-1/2})$. Then $\\sqrt{n}(\\hat\\mu_R - \\mu)$ has asymptotic variance $\\operatorname{Var}(Y_i(1)) + R(g_*)$, which is at most the asymptotic variance $\\operatorname{Var}(Y_i(1)) + R(0)$ of $\\sqrt{n}(\\hat\\mu - \\mu)$.",
    "problem": "Note that the Horvitz--Thompson estimator is the adjusted estimator $\\hat\\mu_R$ with the zero function in place of $\\hat g_0$ and $\\hat g_1$. For a fixed function $g$, the summand of $\\hat\\mu_R$ has variance $\\operatorname{Var}(Y_i(1)) + R(g)$, where\n\\[\nR(g) = \\mathrm{E}\\biggl[ \\frac{1 - \\pi(W_i)}{\\pi(W_i)}\\bigl( Y_i(1) - g(W_i) \\bigr)^2 \\biggr],\n\\]\nand the printed weights make the population weighted least-squares criterion on the treated units proportional to $R(g)$. The fit minimizes $R$ only over the family of functions in which the two regressions are fitted, which I call the working family. The comparison with the Horvitz--Thompson estimator requires that the limit $g_*$ of the fit satisfy $R(g_*) \\le R(0)$. This holds when the working family contains the zero function, and it can fail otherwise. Part (c) as printed admits every family with a finite number of parameters. Every linear model with freely fitted coefficients contains the zero function, and the usual working models satisfy this condition.",
    "what_survives": "The sentence on p.~7 that adjustments ``can be constructed so that they cannot harm, but can dramatically improve, asymptotic precision'' asserts that suitable adjustments exist, and it stands. No other result of the paper relies on part (c), and the paper states on p.~10 that part (c) does not apply to its HTLOESS estimator. In the counterexample both estimators remain unbiased and root-$n$ consistent. What changes is the hypothesis of part (c)."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-E002",
   "work": {
    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
   },
   "work_id": "P046"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "exact_counterexample",
   "exhibits": [
    {
     "alt": "Crop of part (d) of Proposition 4 and the first four lines of the next paragraph. Part (d) reads: If g-hat-0 and g-hat-1 are consistent for the true regression function, then mu-hat-R attains the semiparametric efficiency bound. Red outlines surround the words the semi-parametric efficiency bound, which run over a line break. The paragraph reads: All parts of Proposition 4 are previously known results. For completeness, we provide a proof for Proposition 4(a) in the appendix. See van der Vaart (1998) for Proposition 4(b). See Section 3 of Cao et al. (2009) for Proposition 4(c); see also Rubin and van der Laan (2008). See Theorem 5.1 in Chernozhukov et al. (2018) for Proposition 4(d).",
     "caption": "Proposition 4(d) and the paragraph that follows the proposition, p. 8 (PDF p. 7). The outline marks “the semiparametric efficiency bound”; the last line shown cites Theorem 5.1 of Chernozhukov et al. (2018) for part (d).",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E003.png"
    }
   ],
   "formal_release": null,
   "formal_statements": [
    {
     "id": "P046-T9",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-T9"
    }
   ],
   "headline": "“The semiparametric efficiency bound” in Proposition 4(d) is the bound of Hahn (1998) for an unrestricted covariate distribution, and with this meaning the statement holds whether or not the covariate distribution is known.",
   "id": "P046-E003",
   "location": "Proposition 4(d), p. 8 (PDF p. 7), with the set-up on p. 5 (PDF p. 4)",
   "pdf_page": 7,
   "printed_page": "8",
   "reaches": [],
   "related_entries": [],
   "revisions": [],
   "same_defect_as": [],
   "severity": null,
   "subject": "paper",
   "target_ids": [
    "P046-T9"
   ],
   "text": {
    "as_printed": "(d) If $\\hat g_0$ and $\\hat g_1$ are consistent for the true regression function, then $\\hat\\mu_R$ attains \\hl{the semiparametric efficiency bound}.",
    "corrected": "(d) If \\hl{$\\|\\hat g_j - m\\|_{L^2(F_W)} \\to 0$ in probability for $j = 0, 1$, where $m(w) = \\mathrm{E}[Y_i(1) \\mid W_i = w]$}, then $\\hat\\mu_R$ attains \\hl{the efficiency bound of Hahn (1998) for $\\mu$ in the model with a known propensity score and an unrestricted covariate distribution $F_W$, which equals $\\mathrm{E}[m(W_i)(1 - m(W_i))/\\pi(W_i)] + \\operatorname{Var}(m(W_i))$}.",
    "intended_meaning": "The paragraph after Proposition 4 cites Theorem 5.1 of Chernozhukov et al.\\ (2018) for part (d). That theorem, as it appears in the preprint arXiv:1608.00060v6, names the efficiency bound of Hahn (1998), which is the bound in the model with an unrestricted distribution $F_W$ of the covariate. Let $m(w) = \\mathrm{E}[Y_i(1) \\mid W_i = w]$. For $\\mu$ and a known propensity score this bound equals $\\mathrm{E}[m(W_i)(1 - m(W_i))/\\pi(W_i)] + \\operatorname{Var}(m(W_i))$, and $\\hat\\mu_R$ attains it when $\\hat g_0$ and $\\hat g_1$ converge to $m$ in $L^2(F_W)$, whether or not $F_W$ is known. The set-up on p.~5 allows $F_W$ to be ``either known or unknown''. In the smaller model in which $F_W$ is known the bound is lower by $\\operatorname{Var}(m(W_i))$, and part (d) does not claim that $\\hat\\mu_R$ attains that lower bound."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P046/#P046-E003",
   "work": {
    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
   },
   "work_id": "P046"
  },
  {
   "before_review": null,
   "class": "clarity_note",
   "date_modified": "2026-10-25",
   "date_published": "2026-10-25",
   "evidence_type": "complete_derivation",
   "exhibits": [
    {
     "alt": "Crop of the last six lines of page 9. A red outline surrounds the sentence: The second estimator, labelled LOESS, uses an LOESS regression for the predictions. The text continues: We use LOESS regression with locally linear fits and the smoothing parameter, alpha, is set to 5 times N to the power minus one third, where N is the number of observations used in the regression. In this case, N is approximately n/2, because this is the number of observations with X-i equal to 1.",
     "caption": "Section 3.2, p. 9 (PDF p. 8). The outline marks the sentence that introduces the LOESS estimator; the lines after it give the degree and span of the fits and no rule for missing predictions.",
     "url": "https://pmaronow.github.io/llm-audit/works/P046/exhibits/P046-E005.png"
    },
    {
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   "headline": "The simulation code applies two rules that Section 3.2 does not state: a missing LOESS prediction is set to $0.5$, and an estimated propensity score below $0.01$ is set to $0.01$.",
   "id": "P046-E005",
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    "as_printed": "\\emph{p.~9.} The second estimator, labelled \\texttt{LOESS}, uses an LOESS regression for the predictions. We use LOESS regression with locally linear fits and the smoothing parameter, $\\alpha$, is set to $5N^{-1/3}$, where $N$ is the number of observations used in the regression.\n\n\\emph{p.~10.} The first of these estimators, labelled \\texttt{DRLOG}, uses ordinary linear logistic regression to construct both $\\hat g(W_i)$ and $\\hat\\pi(W_i)$. The second estimator, labelled \\texttt{DRLOESS}, instead uses LOESS regression for both functions. We use the same type of LOESS regression here as above.",
    "corrected": "Add to Section 3.2: \\hl{For every LOESS fit, missing predictions are replaced by $0.5$. Estimated propensity scores used in \\texttt{DRLOG} and \\texttt{DRLOESS} are floored at $0.01$ from below only, with no upper cap.}",
    "intended_meaning": "The passages name the regression families, the degree and span of the LOESS fits and the estimator formulas, and nothing in them is false. They do not fully specify the estimators behind Figure 3. In the replication code (\\texttt{inesims.R}), every LOESS prediction that is missing is replaced by $0.5$, which enters \\texttt{LOESS}, \\texttt{DRLOESS} and \\texttt{HTLOESS}, and the estimated propensity scores of \\texttt{DRLOG} and \\texttt{DRLOESS} are raised to $0.01$ when they fall below it, with no upper cap. Both rules can take effect. Their effect on the reported RMSEs was not measured, and no reported value is shown to change."
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    "arxiv_id": "2108.11342",
    "doi": "10.1353/obs.2025.a956837",
    "title": "Nonparametric identification is not enough, but randomized controlled trials are",
    "url": "https://pmaronow.github.io/llm-audit/works/P046/"
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     "body": "<p>Proposition 1, p. 6 (PDF p. 5). Every measurable observed-data estimator sequence has a positive error threshold with failure probability bounded away from zero along arbitrarily large sample sizes and valid population laws, so uniform consistency over the unrestricted class fails.</p>",
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     "body": "Proposition 1, p. 6 (PDF p. 5). Every measurable observed-data estimator sequence has a positive error threshold with failure probability bounded away from zero along arbitrarily large sample sizes and valid population laws, so uniform consistency over the unrestricted class fails.",
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 "reproduction": {
  "agreement": "agrees",
  "status": "reduced_simulation",
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    "agreement": "to_rounding",
    "location": "journal pp. 8-9 (PDF pp. 7-8), Figure 2 on p. 9",
    "method": "I integrated the three data-generating formulas and checked their extrema and the supplied functions. Each mean is $1/2$, and the extrema agree with the stated approximate $0.1$ and $0.9$; the third panel is schematic.",
    "status": "recalculated",
    "target": "Figure 2 and DGP calibration"
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    "agreement": "within_mc_error",
    "location": "journal p. 10 (PDF p. 9), left panel; discussion pp. 10-11 (PDF pp. 9-10)",
    "method": "I regenerated all five plotted sample sizes from the authors’ saved 10,000-draw results and compared the endpoints visually. I also ran the authors’ code at $n=1000$ and $n=10{,}000$ with 2000 draws per cell; the RMSEs agree with the saved results within Monte Carlo error.",
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    "target": "Figure 3, DGP 1"
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    "agreement": "within_mc_error",
    "location": "journal p. 10 (PDF p. 9), middle panel; discussion p. 11 (PDF p. 10)",
    "method": "I regenerated all five plotted sample sizes from the authors’ saved 10,000-draw results and compared the endpoints visually. I also ran the authors’ code at $n=1000$ and $n=10{,}000$ with 2000 draws per cell; the RMSEs agree with the saved results within Monte Carlo error.",
    "status": "reproduced_reduced",
    "target": "Figure 3, DGP 2"
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    "agreement": "within_mc_error",
    "location": "journal p. 10 (PDF p. 9), right panel; discussion p. 11 (PDF p. 10)",
    "method": "I regenerated all five plotted sample sizes from the authors’ saved 10,000-draw results and compared the endpoints visually. I also ran the authors’ code at $n=1000$ and $n=10{,}000$ with 2000 draws per cell; the RMSEs agree with the saved results within Monte Carlo error.",
    "status": "reproduced_reduced",
    "target": "Figure 3, DGP 3"
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 "summary": "The paper argues that randomized controlled trials are special among studies with unconfounded treatment. With a continuous confounder and an unknown propensity score, no uniformly consistent estimator of the average treatment effect exists without further assumptions. With a known propensity score, the paper presents unbiased, root-$n$ consistent estimators, which a simulation illustrates, and a finite-sample confidence interval. 3 entries are classified as errors (1 Moderate, 2 Minor), and 3 are clarity notes; one possible citation error remains unresolved. The Moderate error, \\fid{P046-E001}, is that the interval of Proposition 3 is not valid at every level. A wider interval is valid.",
 "what_was_checked": "I checked the following points and found no error in them.\n\\begin{itemize}\n\\item Proposition 1. The impossibility result for the model without further assumptions was rebuilt by two separate arguments for a lower bound, since the paper has no proof of it.\n\\item Proposition 2: the unbiasedness of the Horvitz--Thompson estimator, the variance bound $1/(n\\delta)$ with the uniform root-$n$ rate over populations with a common positivity constant $\\delta>0$, and asymptotic normality for each fixed population.\n\\item The algebra of the exponent in the proof of Proposition 3. The centring of the bound is the subject of \\fid{P046-E001}.\n\\item The proof of Proposition 4(a), equality by equality, and parts (b) and (d) by a direct calculation of the conditional mean and variance of the summands, with part (d) read as in \\fid{P046-E003}. Part (c) is the subject of \\fid{P046-E002}.\n\\item The three data generating processes. Each has mean $1/2$, the stated minima and maxima of the propensity score are approximately right, and Figure 2 and the code agree with the formulas.\n\\item The code against Section 3: the six estimators, the LOESS span and degree, the cross-fitting on opposite halves, the five sample sizes, and 10,000 replications in each of the 15 result files in the replication files. Two rules that the paper does not describe are the subject of \\fid{P046-E005}.\n\\item Figure 3 and Section 3.3. The figure was redrawn from the 15 result files and agrees with the published one within the precision of reading a graph, and the vertical shifts in the plotting script are the ones disclosed on p.~11. Two cells ($n = 1000$ under the second and third data generating processes) were recomputed with the authors' code and seed and agree with the stored bias and RMSE to about $10^{-15}$. A smaller new simulation ($n = 1000$ and $n = 10{,}000$, with 2000 draws per cell) agrees with the stored RMSEs within Monte Carlo error. Every statement of Section 3.3 about orderings and rates agrees with the stored values.\n\\end{itemize}\nThe following limits apply to my checks.\n\\begin{itemize}\n\\item The full simulation (five sample sizes, 10,000 draws per cell) was not repeated. The sample sizes 3162, 31,623 and 100,000 were compared from the result files only. Figure 3 prints no numbers, and its published values are readings from the graph.\n\\item Whether the interval of Proposition 3 as printed is valid at conventional levels in every design.\n\\item I checked three source pointers against publicly available passages and found nothing wrong: the original Robins--Wasserman post, search-indexed excerpts of Cao et al.'s author manuscript and the Chernozhukov preprint. The article of Robins and Ritov (1997) could not be obtained, and the attribution of Proposition 1 to its Theorem 3 was not checked. The version of record of Chernozhukov et al.\\ was not read.\n\\item The bibliographic details of the reference list were not compared with the sources.\n\\end{itemize}\nI do not infer that the rest of the paper is correct from an absence of corrections.",
 "work": {
  "arxiv_id": "2108.11342",
  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "James M. Robins",
   "Theo Saarinen",
   "Fredrik Sävje",
   "Jasjeet Sekhon"
  ],
  "category": "Essays, Commentary, Reports",
  "citation": "P. M. Aronow, James M. Robins, Theo Saarinen, Fredrik Sävje, and Jasjeet Sekhon (2025). Nonparametric identification is not enough, but randomized controlled trials are. Observational Studies 11(1):3–16.",
  "doi": "10.1353/obs.2025.a956837",
  "edition": {
   "pages_in_audited_file": 15,
   "sha256": "296af4ee19242f4cb3360da23d768c7af9c59e8c778ab1a832aa209ce6d317c3",
   "source_url": "https://doi.org/10.1353/obs.2025.a956837",
   "venue": "Observational Studies",
   "year": "2025"
  },
  "id": "P046",
  "later_materials": null,
  "title": "Nonparametric identification is not enough, but randomized controlled trials are",
  "type": "Commentary/article",
  "url": "https://pmaronow.github.io/llm-audit/works/P046/",
  "venue": "Observational Studies",
  "year": "2025"
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