{
 "claims": [
  {
   "claim": "Mean ignorability within degree classes, with every class sampled with positive probability (Condition 1), together with knowledge of the degree-specific sampling probability up to an unknown scale (Condition 2), nonparametrically identifies the population mean by the normalized inverse-probability formula (2).",
   "claim_id": "P021-H1",
   "related_corrections": [
    "P021-E001",
    "P021-N003"
   ],
   "tied_to": "Conditions 1 and 2; Proposition 1; Eqs. (2)-(4)"
  },
  {
   "claim": "Under the convergence setup of Section 2 and Conditions 1 and 2 with $f(k)$ proportional to $k$, the Volz–Heckathorn estimator (1) is consistent for the population mean.",
   "claim_id": "P021-H2",
   "related_corrections": [
    "P021-E001"
   ],
   "tied_to": "Corollary 1; Eq. (1); Proposition 1"
  },
  {
   "claim": "The conditions for consistency of the VH estimator are weaker than previously assumed: they are less stringent than the traditional random-walk model, consistency requires neither random sampling nor an actual network, and neither Condition requires any of the network-structure assumptions of Table 1.",
   "claim_id": "P021-H3",
   "related_corrections": [],
   "tied_to": "Corollary 1 and Conditions 1-2 set against Table 1"
  },
  {
   "claim": "Ignorability given degree alone does not identify the population mean in RDS because the population degree distribution is unknown; specifying the conditional sampling probability up to scale (Condition 2) supplies the missing reweighting, which is the difficulty RDS adds to other non-probability samples.",
   "claim_id": "P021-H4",
   "related_corrections": [],
   "tied_to": "Conditions 1 and 2; Proposition 1, Eq. (3)"
  },
  {
   "claim": "Without information on unsampled subjects neither Condition 1 nor Condition 2 has directly testable implications, so the value of any diagnostic depends on further assumptions about the generative (recruitment) process.",
   "claim_id": "P021-H5",
   "related_corrections": [
    "P021-N003"
   ],
   "tied_to": "Conditions 1 and 2; Proposition 1; Corollary 1; Table 1"
  },
  {
   "claim": "The VH estimator and its inverse-probability variants may be appropriate even when diagnostics predicated on a more restrictive network model, such as those of Gile et al. (2015), fail.",
   "claim_id": "P021-H6",
   "related_corrections": [],
   "tied_to": "Table 1; Proposition 1; Corollary 1"
  }
 ],
 "date_published": "2026-10-25",
 "findings": [
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     "caption": "Section 2, first paragraph, p. 101 (PDF p. 2). The outline marks the words “Without loss of generality”, which introduce the bounds on the outcome and on the degree.",
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   ],
   "headline": "The bounds on the outcome and on the degree are assumptions that hold for every population in the sequence, and they are not made without loss of generality.",
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   "location": "Section 2, first paragraph, last sentence, p. 101 (PDF p. 2)",
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   "text": {
    "as_printed": "Formally, consider a sequence of populations and samples converging weakly to a joint limit distribution on the outcome, (reported) degree, and sample, denoted $(Y, D, S)$. Let the $\\mathrm{E}[\\cdot]$ and $\\Pr[\\cdot]$ operators refer to features of this limiting distribution. In RDS, we observe the empirical joint distribution of the outcome $Y$ and degree $D$ conditional on the sampling indicator $S = 1$. \\hl{Without loss of generality}, suppose that $Y$ has bounded support and that $D$ has support in the set $\\{1, \\ldots, K\\}$.",
    "corrected": "Formally, consider a sequence of populations and samples \\hl{whose empirical joint laws converge weakly, in probability if they are random,} to a joint limit distribution on the outcome, (reported) degree, and sample, denoted $(Y, D, S)$. Let the $\\mathrm{E}[\\cdot]$ and $\\Pr[\\cdot]$ operators refer to features of this limiting distribution. In RDS, we observe the empirical joint distribution of the outcome $Y$ and degree $D$ conditional on the sampling indicator $S = 1$. \\hl{For simplicity, assume that there exist constants $B<\\infty$ and $K<\\infty$ such that every population in the sequence has outcomes in $[-B,B]$ and degrees in $\\{1, \\ldots, K\\}$.}",
    "intended_meaning": "The two bounds are meant as standing assumptions, made for simplicity. Every population in the sequence has outcomes in a common interval $[-B,B]$ and degrees in $\\{1,\\ldots,K\\}$. On this reading, Proposition 1 and Corollary 1 are correct. For Corollary 1, suppose that $\\Pr[S=1\\mid D=k]=ak$ with $a>0$, and let $A_N$ and $H_N$ denote the averages of $sy/d$ and $s/d$ over the $N$ units of a population, where $s=1$ for a sampled unit and $s=0$ otherwise. The estimator (1) equals $A_N/H_N$. Since the functions $sy/d$ and $s/d$ are bounded and continuous on $[-B,B]\\times\\{1,\\ldots,K\\}\\times\\{0,1\\}$, the weak convergence assumed in the first sentence of Section 2 implies that\n\\[\nH_N \\to \\mathrm{E}[S/D] = a, \\qquad A_N \\to \\mathrm{E}[SY/D] = a\\,\\mathrm{E}[Y],\n\\]\nwhere I used Condition 1 in the last equality. I conclude that $A_N/H_N \\to \\mathrm{E}[Y]$.\n\nThe words ``without loss of generality'' claim more than this, and two exact examples explain the outcome restriction. First, let $D=1$, let $\\Pr[Y=j]=2/3^{j+1}$ for every integer $j\\ge 0$, and let $S$ be independent of $Y$ with $\\Pr[S=1]=2/5$. Then Conditions 1 and 2 hold with $f(1)=1$ and $c=2/5$, and $\\mathrm{E}[Y]=1/2$. The outcome is unbounded, and clipping $Y$ at an integer $M\\ge 1$ changes the mean to $(1-3^{-M})/2$. Second, a bound on the limiting $Y$ alone is not enough for Corollary 1. For each $m\\ge 1$, I consider a population of $N=3m$ units of degree $1$ in which $m$ units are sampled, one sampled unit has the outcome $2m$, and every other unit has the outcome $0$. The empirical joint laws converge weakly to the law with mass $2/3$ at $(Y,D,S)=(0,1,0)$ and mass $1/3$ at $(0,1,1)$. This limit satisfies Conditions 1 and 2 with $f(1)=1$ and $c=1/3$, and it has $\\mathrm{E}[Y]=0$. However, $A_N=2/3$ and $H_N=1/3$, and the estimator (1) equals $2$ for every $m$.\n\nThe common bound on the degree is stronger than Corollary 1 needs. When $\\Pr[S=1\\mid D=k]=ak$, the positivity in Condition 1 excludes degree $0$, and the inequality $ak\\le 1$ bounds the degrees of the limit. A condition under which $A_N$ and $H_N$ converge to $\\mathrm{E}[SY/D]$ and $\\mathrm{E}[S/D]$ can replace the common bound on the outcome."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P021/#P021-E001",
   "work": {
    "arxiv_id": "1504.03574",
    "doi": "10.1016/j.spl.2015.07.003",
    "title": "Nonparametric Identification for Respondent-Driven Sampling",
    "url": "https://pmaronow.github.io/llm-audit/works/P021/"
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     "caption": "References, p. 102 (PDF p. 3). The outlines mark the two arXiv identifiers printed without the dot. The entry for Crawford (2014), three lines below the first, prints its identifier with the dot.",
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   "headline": "The two arXiv identifiers should read arXiv:1304.3505 and arXiv:1401.4718; each is printed without the dot.",
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   "location": "References, p. 102 (PDF p. 3): the entries for Berchenko, Rosenblatt and Frost (2013) and for Lunagomez and Airoldi (2014)",
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   "text": {
    "as_printed": "Berchenko, Y., Rosenblatt, J., Frost, S.D., 2013, Modeling and analysing respondent driven sampling as a counting process. arXiv preprint arXiv:\\hl{13043505}.\n\nLunagomez, S., Airoldi, E.M., 2014, Valid inference from non-ignorable network sampling designs. arXiv preprint arXiv:\\hl{14014718}.",
    "corrected": "Berchenko, Y., Rosenblatt, J., Frost, S.D., 2013, Modeling and analysing respondent driven sampling as a counting process. arXiv preprint arXiv:\\hl{1304.3505}.\n\nLunagomez, S., Airoldi, E.M., 2014, Valid inference from non-ignorable network sampling designs. arXiv preprint arXiv:\\hl{1401.4718}.\n\nThe links should point to \\url{https://arxiv.org/abs/1304.3505} and \\url{https://arxiv.org/abs/1401.4718}.",
    "evidence": "I checked the page: both identifiers appear without the dot on p.~102, while the entry for Crawford (2014) on the same page prints 1406.0721 with the dot. In the PDF, the two links point to addresses that end in \\texttt{13043505} and \\texttt{14014718}. The first versions of arXiv:1304.3505 and arXiv:1401.4718 carry the cited titles and authors.",
    "problem": "An arXiv identifier of 2013 or 2014 has a dot between the four digits of the year and month and the serial number. Both entries print the eight digits without the dot, and the links in the PDF point to the same strings.",
    "what_survives": "No result, proof or estimator depends on these entries. Only the two identifiers and their links change."
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   "work": {
    "arxiv_id": "1504.03574",
    "doi": "10.1016/j.spl.2015.07.003",
    "title": "Nonparametric Identification for Respondent-Driven Sampling",
    "url": "https://pmaronow.github.io/llm-audit/works/P021/"
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   "headline": "Two reference entries give a supplement as the ordinary issue 1: the issues are AIDS Behav. 12 (Suppl 1) and J. Urban Health 83 (Suppl 1).",
   "id": "P021-N001",
   "location": "References, p. 102 (PDF p. 3): the entries for Johnston et al. (2008) and for Salganik (2006)",
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   "text": {
    "as_printed": "Johnston, L.G., Malekinejad, M., Kendall, C., Iuppa, I.M., Rutherford, G.W., 2008. Implementation challenges to using respondent-driven sampling methodology for HIV biological and behavioral surveillance: field experiences in international settings. AIDS Behav. 12 \\hl{(1)}, 131--141.\n\nSalganik, M.J., 2006. Variance estimation, design effects, and sample size calculations for respondent-driven sampling. J. Urban Health 83 \\hl{(1)}, 98--112.",
    "corrected": "Johnston, L.G., Malekinejad, M., Kendall, C., Iuppa, I.M., Rutherford, G.W., 2008. Implementation challenges to using respondent-driven sampling methodology for HIV biological and behavioral surveillance: field experiences in international settings. AIDS Behav. 12 \\hl{(Suppl 1)}, 131--141.\n\nSalganik, M.J., 2006. Variance estimation, design effects, and sample size calculations for respondent-driven sampling. J. Urban Health 83 \\hl{(Suppl 1)}, 98--112.",
    "evidence": "I compared the entries with the publisher's pages for the two articles. The page for the article of Johnston et al.\\ gives ``AIDS Behav 12 (Suppl 1), 131--141 (2008)'' (\\url{https://link.springer.com/article/10.1007/s10461-008-9413-1}). The page for the article of Salganik gives ``J Urban Health 83 (Suppl 1), 98--112 (2006)'' (\\url{https://link.springer.com/article/10.1007/s11524-006-9106-x}). The National Library of Medicine indexes the same articles as 12(4 Suppl):S131--S141 and 83(6 Suppl):i98--i112.",
    "problem": "Both articles appeared in supplements to their volumes, and the two entries print the supplement as the ordinary issue ``(1)''. The volumes and the page ranges are correct, and both articles can still be found from them. The two arXiv identifiers in the same list are the subject of \\fid{P021-E002}.",
    "what_survives": "No result depends on these entries. A comparison with the article of Gile et al.\\ (2015) supports the attributions to it. The phrase quoted with the locator ``pg.~3'' is on the third page of their article, p.~243, and Table 1 keeps every substantive entry of their Table 1 on p.~244. The other entries of the reference list agree with the publishers' or indexers' records in volume, issue, year and first page. The entries for Broadhead et al.\\ (1998) and McCreesh et al.\\ (2012) give only the first page of the ranges 42--57 and 138--147."
   },
   "url": "https://pmaronow.github.io/llm-audit/works/P021/#P021-N001",
   "work": {
    "arxiv_id": "1504.03574",
    "doi": "10.1016/j.spl.2015.07.003",
    "title": "Nonparametric Identification for Respondent-Driven Sampling",
    "url": "https://pmaronow.github.io/llm-audit/works/P021/"
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   "exhibits": [
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     "alt": "Crop of Condition 1, Condition 2, Proposition 1 with equation (2), and the first line of the proof. Red outlines surround three items: the equation Pr[S = 1 | D = k] = c f(k) in Condition 2, and the summation signs with limits k = 1 and K in the numerator and in the denominator of equation (2). Condition 1, which is not marked, begins: For all k such that Pr[D = k] > 0.",
     "caption": "Conditions 1 and 2 and Proposition 1, p. 101 (PDF p. 2). The outlines mark the equation of Condition 2, which carries no restriction on $k$, and the two sums in (2), which run from $k=1$ to $K$.",
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   "headline": "Condition 2 and the sums in (2) and (4) are meant for the degrees $k$ such that $\\Pr[D=k]>0$, the restriction that Condition 1 states.",
   "id": "P021-N003",
   "location": "Condition 2, Proposition 1 with equation (2), and the proof with equations (3) and (4), p. 101 (PDF p. 2); Discussion, second paragraph, p. 102 (PDF p. 3)",
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   "text": {
    "as_printed": "\\emph{Section 2, p.~101.} Condition 2 (Knowledge of the Conditional Probability of Sampling). $\\Pr[S = 1 \\mid D = \\hl{k}] = cf(k)$, where $f(\\cdot)$ is known and $c$ is an unknown scale parameter.\n\nProposition 1. Given Conditions 1 and 2, the population mean is identified, with\n\\[\n\\mathrm{E}[Y] = \\frac{\\hl{\\sum_{k=1}^{K}} \\mathrm{E}[Y \\mid S = 1, D = k]\\,\\frac{\\Pr[D = k \\mid S = 1]}{f(k)}}{\\hl{\\sum_{k=1}^{K}} \\frac{\\Pr[D = k \\mid S = 1]}{f(k)}}. \\qquad (2)\n\\]\n\n\\emph{Discussion, p.~102.} Without knowledge of the characteristics of the unsampled subjects, \\hl{neither Condition 1 nor Condition 2 has directly testable implications}, and thus the value of any diagnostics must depend on further assumptions about the generative process.",
    "corrected": "\\emph{Section 2, p.~101.} \\hl{Let $A=\\{k:\\Pr[D=k]>0\\}$.} Condition 2 (Knowledge of the Conditional Probability of Sampling). \\hl{For all $k\\in A$,} $\\Pr[S = 1 \\mid D = k] = cf(k)$, where $f(\\cdot)$ is known and $c$ is an unknown scale parameter.\n\nProposition 1. Given Conditions 1 and 2, the population mean is identified, with\n\\[\n\\mathrm{E}[Y] = \\frac{\\hl{\\sum_{k\\in A}} \\mathrm{E}[Y \\mid S = 1, D = k]\\,\\frac{\\Pr[D = k \\mid S = 1]}{f(k)}}{\\hl{\\sum_{k\\in A}} \\frac{\\Pr[D = k \\mid S = 1]}{f(k)}}. \\qquad (2)\n\\]\nThe proof's ``for each degree $k$'' and equation (3) refer to $k\\in A$, and both sums in each fraction of (4) are over $k\\in A$. Under Condition 1, $A=\\{k:\\Pr[D=k\\mid S=1]>0\\}$, so the observed distribution determines $A$.\n\n\\emph{Discussion, p.~102.} Without knowledge of the characteristics of the unsampled subjects, \\hl{and apart from compatibility of the observed degree support with the known sampling-probability profile,} neither Condition 1 nor Condition 2 has directly testable implications, and thus the value of any diagnostics must depend on further assumptions about the generative process.",
    "intended_meaning": "Condition 1 is stated for the degrees $k$ such that $\\Pr[D=k]>0$. Condition 2, the proof's ``for each degree $k$'', equation (3) and the sums in (2) and (4) are meant for the same degrees, and I write $A=\\{k:\\Pr[D=k]>0\\}$ for this set. Read literally, the sums also include every $k\\le K$ such that $\\Pr[D=k]=0$. For such $k$ the conditional means are not defined, and the weight $\\Pr[D=k\\mid S=1]/f(k)$ is $0/0$ when $f(k)=0$. For example, let $K=3$, $\\Pr[D=1]=\\Pr[D=3]=1/2$, $f(1)=1$, $f(2)=0$, $f(3)=3$ and $c=1/4$, with $Y=0$ when $D=1$ and $Y=1$ when $D=3$. Then $\\Pr[S=1]=1/2$, the observed degree probabilities are $(1/4,0,3/4)$, and the term $k=2$ of (2) is $0/0$. With the sums taken over $A=\\{1,3\\}$, I obtain $(1/4)/(1/2)=1/2$, which is $\\mathrm{E}[Y]$.\n\nOn $A$ the proof is correct and needs no further assumption. By Condition 1, $\\Pr[S=1\\mid D=k]>0$ for all $k\\in A$. It follows that $\\Pr[S=1]>0$, that $A=\\{k:\\Pr[D=k\\mid S=1]>0\\}$, and that $cf(k)>0$ for all $k\\in A$. By Bayes' rule and Condition 2, I have\n\\[\n\\frac{\\Pr[D=k\\mid S=1]}{f(k)} = \\frac{c}{\\Pr[S=1]}\\,\\Pr[D=k] \\qquad \\text{for all } k\\in A.\n\\]\nSumming over $A$, I find that the denominator of (2) equals $c/\\Pr[S=1]$ and, by Condition 1, that the numerator equals $(c/\\Pr[S=1])\\,\\mathrm{E}[Y]$. Their ratio is $\\mathrm{E}[Y]$.\n\nThe sentence of the Discussion needs a qualification of the same kind. If the known function $f$ vanishes at some $k$, then Conditions 1 and 2 imply that $\\Pr[D=k\\mid S=1]=0$, which can be checked in the sample. For an $f$ that is strictly positive on the observed degrees, the two conditions place no further restriction on the observed law, and the sentence holds as printed. In Corollary 1, $f(k)\\propto k$ has no zero on $\\{1,\\ldots,K\\}$."
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   "work": {
    "arxiv_id": "1504.03574",
    "doi": "10.1016/j.spl.2015.07.003",
    "title": "Nonparametric Identification for Respondent-Driven Sampling",
    "url": "https://pmaronow.github.io/llm-audit/works/P021/"
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     "body": "<p>Proposition 1 and Eqs. (2)–(4), p. 101 (PDF p. 2). Mean ignorability and sampling probabilities known up to scale identify the population mean by the observed inverse-profile ratio, with the printed Bayes and cancellation equalities, summing over positive-mass degrees under the standing support convention.</p>",
     "headline": "Proposition 1, with Eq. (2) and the printed proof, Eqs. (3)–(4)",
     "location": "p. 101 (PDF p. 2)"
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     "body": "Proposition 1 and Eqs. (2)–(4), p. 101 (PDF p. 2). Mean ignorability and sampling probabilities known up to scale identify the population mean by the observed inverse-profile ratio, with the printed Bayes and cancellation equalities, summing over positive-mass degrees under the standing support convention.",
     "headline": "Proposition 1, with Eq. (2) and the printed proof, Eqs. (3)–(4)",
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      "credit_reason": "{'source': 'adjudication', 'decision': 'rejected', 'what': \"At P021/P021_RDS.lean:729, corollary_one concludes convergence in probability of the sampled-unit VH ratio defined at line 619 to populationMean F, matching 'the VH estimator is consistent for E[Y]'. The common bounds in RDSState (line 21) and the realized-empirical convergence premise (line 686) are explicit in the supplied standing assumptions and claim definitions. Testing every continuous real function on this compact state space expresses the specified weak convergence, and Mathlib's TendstoInMeasure under a probability measure expresses the specified consistency. The proportional profile a * degreeValue k matches f(k) proportional to k, and the explicit B >= 0, K > 0, and a != 0 hypotheses exclude no admissible cases. Assigning zero to an empty sample loses no consistency conclusion, since the same limiting conditions force a positive inverse-degree denominator and the empty-sample probability tends to zero (line 911).\"}",
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     "body": "<p>Corollary 1, p. 101 (PDF p. 2), and Eq. (1), p. 100 (PDF p. 1). Under the common bounded support and weak convergence of the realized joint empirical laws, mean ignorability and sampling probabilities proportional to degree imply that the VH estimator converges to the population outcome mean, in probability for random populations and ordinarily for deterministic populations. The sample consists of the units marked S = 1. Equation (1) is assigned zero on an empty sample; finite stages may have empty samples.</p>",
     "headline": "Corollary 1, with the sentence introducing it and the VH estimator, Eq. (1)",
     "location": "p. 101 (PDF p. 2); Eq. (1) is on p. 100 (PDF p. 1)"
    },
    "id": "P021-T2",
    "text": {
     "body": "Corollary 1, p. 101 (PDF p. 2), and Eq. (1), p. 100 (PDF p. 1). Under the common bounded support and weak convergence of the realized joint empirical laws, mean ignorability and sampling probabilities proportional to degree imply that the VH estimator converges to the population outcome mean, in probability for random populations and ordinarily for deterministic populations. The sample consists of the units marked S = 1. Equation (1) is assigned zero on an empty sample; finite stages may have empty samples.",
     "headline": "Corollary 1, with the sentence introducing it and the VH estimator, Eq. (1)",
     "location": "p. 101 (PDF p. 2); Eq. (1) is on p. 100 (PDF p. 1)"
    },
    "url": "https://pmaronow.github.io/llm-audit/works/P021/#P021-T2"
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 "summary": "This note asks when the Volz--Heckathorn estimator for respondent-driven sampling is consistent. It shows that the population mean is identified when outcomes are mean-ignorable given degree and the sampling probability is positive and is a function of degree known up to scale (Proposition 1), and that the estimator is consistent under the standing convergence and common-bound assumptions when this function is proportional to degree (Corollary 1). 2 entries are classified as errors (2 Minor), both in the reference list, and 2 are clarity notes on how the assumptions are stated. Two possible errors were not confirmed.",
 "what_was_checked": "I checked the following, with the qualifications stated below.\n\\begin{itemize}\n\\item Equation (1) is the inverse-degree ratio estimator of Volz and Heckathorn, and it is the sample version of (2) with $f(k)=k$.\n\\item Proposition 1 and its proof. Equation (3) follows from Bayes' rule and Condition 2. In (4), the law of total expectation is applied correctly, and the unknown scale $\\Pr[S=1]/c$ cancels. Both steps hold on the degrees with positive probability (see \\fid{P021-N003}). Exact calculations agree with (2) in 25 examples, on 200 randomly generated joint laws, and symbolically for $K=3$.\n\\item Mean ignorability within each degree class is enough for Proposition 1. Formula (2) holds on laws in which sampling depends on the outcome within a degree class, and formula (2) does not change when $f$ is rescaled.\n\\item Corollary 1, under the convergence assumed in the first sentence of Section 2 and bounds common to the sequence (see \\fid{P021-E001}). The derivation does not use independence or the random-walk model.\n\\item The comparison with the random-walk model, for the limiting population law and the limiting law of the visit records. These two laws can be represented by a joint law that satisfies Conditions 1 and 2 with $f(k)=k$. The paper does not define the indicator $S$ when a subject is recruited more than once.\n\\item Three remarks of the Discussion. Ignorability given degree alone does not identify the mean, since two models with the same observed law have means $1/2$ and $1/4$. For a strictly positive $f$, every observed law of $(Y,D)$ given $S=1$ is compatible with Conditions 1 and 2. Neither condition requires any network-structure assumption of Table 1.\n\\item The citations. Each of the 20 works cited in the text has a reference entry, and each of the 20 entries is cited. The quotation from Gile et al.\\ (2015) and the substantive entries of Table 1 agree with that article. The reference entries agree with the publishers' or indexers' records in the checked publication details, apart from the items in \\fid{P021-E002} and \\fid{P021-N001}.\n\\end{itemize}\nI could not obtain the body of the final article of Gile (2011). I read its preprint and the publisher's abstract, which support the citation on p.~100. I compared the other cited works with the note for their publication details only. I did not test on data the remark that the estimator may be appropriate when the diagnostics of Gile et al.\\ (2015) fail; it states a logical possibility. The note has no numerical results, simulations, data or code, and it refers to no appendix or supplement. There were no empirical numerical results to recompute.",
 "work": {
  "arxiv_id": "1504.03574",
  "audit_complete": true,
  "authors": [
   "P. M. Aronow",
   "Forrest W. Crawford"
  ],
  "category": "Articles - Theory and Methodology",
  "citation": "P. M. Aronow and Forrest W. Crawford (2015). Nonparametric Identification for Respondent-Driven Sampling. Statistics and Probability Letters 106:100–102.",
  "doi": "10.1016/j.spl.2015.07.003",
  "edition": {
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   "sha256": "bbe090b4e767b80d2cd4189d70e0d65c238b452f78ef925d1c438732b91db484",
   "source_url": "https://doi.org/10.1016/j.spl.2015.07.003",
   "venue": "Statistics and Probability Letters",
   "year": "2015"
  },
  "id": "P021",
  "later_materials": null,
  "title": "Nonparametric Identification for Respondent-Driven Sampling",
  "type": "Journal article",
  "url": "https://pmaronow.github.io/llm-audit/works/P021/",
  "venue": "Statistics and Probability Letters",
  "year": "2015"
 }
}