Nonparametric Estimation from Incomplete Observations
1958 · paper · Mathematics and statistics
Studies estimating the proportion of items whose lifetimes exceed a given time when some observations are cut short by losses, without assuming a distribution form. Presents the product-limit estimate, which maximizes the likelihood, and discusses its mean, variance, and alternative estimates.
Checked: PDF from jhanley.biostat.mcgill.ca, fetched 5 Oct 2026. Page numbers are those of that PDF.
5 findings
Substance Below Table 463 the text reads "The relation n_j′ ≤ n(u_j) ≤ n_{j+1} holds."; by the paper's own scheme n_{j+1} = n_j′ − λ_j, so the relation should read n_{j+1} ≤ n(u_j) ≤ n_j′.
- p. 8 Paragraph after Table 463, last sentence
The relation n
- p. 8 Section 2.1, sentence after Table 463
The relation n_j' ≤ n(u_j) ≤ n_{j+1} holds.
- p. 8 text under Table 463
The relation n_j′ ≤ n(u_j) ≤ n_{j+1} holds.
Judge's note: Read p.8 page image: both signs are ≤ and the order is n_j′, n(u_j), n_{j+1}. Table 463 places λ_j between n_j′ and n_{j+1}, and p.10 states λ_j = n_j′ − n_{j+1}. Table 464 (p.9) has n_1′ = 7 and n_2 = 5; Table 465 has n_2′ = 92 and n_3 = 72. The printed chain would require 7 ≤ 5. Entrances counted as negative losses are introduced only two paragraphs later and do not make the printed direction general. No explanation found elsewhere in the paper.
Second check: another model agreed
Report a problem with this finding
Substance Table 465, last row (interval 4–5.0, j=7), prints effective sample size N̂_E = "51"; formula (2j) with variance (2f) and the table's own n_j, δ_j gives about 63, and 51 is below the lower bound n(t)/P̂(t) of (2k).
- p. 10 Table 465, row j=7, column N_Ej
16/31 .27 51
- p. 12 Eq. (2j)-(2k)
It can be shown that
- p. 12 Section 2.2, eq. (2j)-(2k) and the sentence after the worked example
Values of N_E are also indicated in Table 465.
- p. 10 Section 2.1, text above Table 465
is the effective sample size defined in (2j) below
Only the authors can say what is right here.
Judge's note: Read Table 465 on p.10 and equations (2f), (2j), (2k) on pp.11-12. The text says N̂_Ej is "the effective sample size defined in (2j) below". Recomputing from the printed n_j and δ_j: the sum of δ_j/[n_j(n_j−δ_j)] over the seven rows is 0.04286, P̂(5.0) = 0.2708, so N̂_E = 62.8 (63.1 with P̂ rounded to .27). The same calculation for row 6 gives 71.8 against the printed 73, and the reviewers' values for the other rows are within a unit or two of print, so the formula is the one used. The (2k) bounds at t = 5.0 are 16/0.27 = 59 and 52/0.73 = 71 (deaths sum to 52; n(t) does not subtract losses at t, per 1.4), and 51 lies outside them. The paper's own worked example on p.12 applies (2j) in exactly this way to Table 464. No statement in the paper explains a different rule for the last row. The intended value (61, 63 or other) is for the authors to say.
Second check: another model agreed
Report a problem with this finding
Reference or label Section 6.2 ends "gives V(μ*) as in (3d).", but no equation is numbered (3d); the V(μ*) display after (3c) in Section 3.1 carries no number.
- p. 23 Section 6.2, last sentence
in (3d).
- p. 14 Section 3.1, unnumbered display after (3c)
is given exactly by
- p. 23 Section 6.2, last sentence
gives V(μ*) as in (3d).
- p. 14 Section 3.1, unnumbered display after (3c)
In Section 6.2 it is shown that the variance of μ* is given exactly by
Judge's note: Read p.23 and all of Section 3 (pp.14-16). Section 3 numbers (3a), (3b) and (3c) only; the display "V(μ*) = 2∫ (1 − P(t))/N⁰(t) ∫ P(u)dudt" on p.14 has no tag at the right margin. The intended target is evident, so the defect is a missing equation number or dangling label.
Second check: another model agreed
Report a problem with this finding
Wording After (4a): "Evidently p̲ ≤ p̂ < p̄." The strict "<" should be "≤": the paper itself later writes "while p̂ = p̄" (Section 4.2) and uses ≤ on both sides in ∏p̲_j ≤ P̂ ≤ ∏p̄_j.
- p. 17 Section 4.1, after (4a)
Evidently
- p. 19 Section 4.2, paragraph after (4k)
within which
- p. 17 Section 4.1, after eq. (4a)
Evidently p ≤ p̂ < p̄.
- p. 19 Section 4.2, paragraph after (4k)
while p̂ = p̄
Judge's note: Read p.17: the first sign has the underbar, the second is a plain <. The same sentence defines p̄ as the value when all deaths precede all losses, a case in which p̂ equals p̄. p.19 says "one can have p(3) < p̲ while p̂ = p̄"; p.18 ends 4.1 with non-strict bounds on both sides; p.17 also has p̲ ≤ p(2) ≤ p(1) ≤ p̄. The strict sign contradicts the paper's own later statements.
Second check: another model agreed
Report a problem with this finding
Wording Likelihood (5a): the first product is printed with lower limit "i=0" (to λ_0), although the loss ages are defined as L_i^(j), "i=1, 2, · · · , λ_j", and the other products run from i=1.
- p. 20 Eq. (5a), first bracket
i=0
- p. 20 Section 5, definition of loss ages
i=1, 2, · · · ,
- p. 20 Section 5, paragraph before (5a)
i=1, 2, · · · , λ_j
- p. 20 Eq. (5a), first bracket
∏ P(L_i^(0)), i=0 to λ_0
Judge's note: Read p.20 page image: the first bracket has i=0 under the product sign, the brackets for λ_1 and λ_k have i=1. The text just above defines i = 1, …, λ_j, and (5c) uses exactly λ_j factors. Starting at 0 would give λ_0 + 1 factors and an undefined L_0^(0). Since (5b) sets P(L_i^(0)) = 1 the result is unaffected.
Second check: another model agreed
Report a problem with this finding
8 minor (typos and the like; most checked by the judge only)
Reference or label Section 2.2 says "The important facts here, derived in Section 6.1, are that P̂(t) is consistent and of negligible bias"; Section 6.1 treats bias and variance only, and consistency is taken up in Section 7, which says "the proof is too lengthy to be given here."
- p. 10 Section 2.2, first sentence
The important facts here, derived in Section 6.1, are that P̂(t) is consistent and of negligible bias
- p. 24 Section 7, first paragraph
However, the proof is too lengthy to be given here.
Judge's note: Read p.10, Section 6.1 on pp.21-22 and Section 7 on pp.24-25. Section 6.1 shows near-unbiasedness (6b) and derives the approximate variance (6c)-(6h); the word consistency does not occur there. Section 7 says a proof of consistency "is in order", that it is too lengthy to give, and treats special cases. The pointer for consistency is therefore to the wrong section and to a result not fully proved in the paper. A loose reading (small bias plus vanishing variance) is possible, hence minor.
Second check: another model agreed
Report a problem with this finding
Wording Section 1.1 says the sample distribution function "equals 1/N times the number of sample values less than the argument t", while Section 1.4 defines P(t) = Pr(T > t) and "F(t) = 1 − P(t)", i.e. values less than or equal to t.
- p. 3 Section 1.1, second paragraph
equals 1/N times the number of sample values less than the argument t
- p. 6 Section 1.4, eq. (1c) and following line
F(t) = 1 − P(t)
Judge's note: Read p.3 and p.6 page images. The 1.1 wording is "less than the argument t"; 1.4 gives P(t) = Pr(T > t), F(t) = 1 − P(t), and says P(t) and its estimate are right-continuous (p.7), so the paper's own convention counts values not exceeding t. The two differ only at the jump points and 1.1 is an informal introduction, so this is a minor consistency point.
Second check: another model agreed
Report a problem with this finding
Typo Section 2.3: "The PL estimate μ̂ of the mean lifetime μ̂ is defined as"; the second symbol should be the unhatted population mean μ.
- p. 12 Section 2.3, first sentence
of the mean lifetime
- p. 12 Section 2.3, first sentence
The PL estimate μ̂ of the mean lifetime μ̂ is defined as
Judge's note: Read p.12 page image: both symbols carry the wide hat. The population mean is written μ without a hat later on the same page ("no satisfactory way to estimate μ") and in Section 1.2 ("the mean life μ").
Second check: none (judge only)
Report a problem with this finding
Typo Section 2.2 worked example: "By (2f) the variance of P(6) is estimated as" should refer to the estimate P̂(6); the display that follows is headed V̂[P̂(6)].
- p. 12 Section 2.2, worked example after (2k)
the variance of P(6)
Judge's note: Read p.12 page image. The preceding sentence has P̂(6) = 21/40 with a hat and the display has V̂[P̂(6)], while the running text prints P(6) with no hat. P(6) is a population constant. A dropped hat, cosmetic.
Second check: none (judge only)
Report a problem with this finding
Typo Section 3.1, foot of p.469: "the estimate ob-" at the line end is followed by "obtained by replacing D⁰(t) by D(t)", giving "ob-obtained".
- p. 14 Section 3.1, last paragraph on the page
the estimate ob-
- p. 14 Section 3.1, last paragraph, last two lines
the estimate ob- obtained by replacing D⁰(t) by D(t)
Judge's note: Read p.14 page image: the second-last line ends "the estimate ob-" and the last line begins with the full word "obtained".
Second check: none (judge only)
Report a problem with this finding
Typo Section 7: "(which exists only for values of t in the interval 0, S)" has no opening bracket for the interval; elsewhere intervals are written with both, e.g. "(0, u)".
- p. 24 Section 7, third paragraph
in the interval 0, S)
- p. 24 Section 7, second paragraph
(which exists only for values of t in the interval 0, S)
Judge's note: Read p.24 page image. The single closing parenthesis has to serve both the interval and the parenthetical aside, so one bracket is missing. Other intervals in the paper, e.g. (0, u_1) on p.7 and (0, u), (u, v) on p.23, have both brackets.
Second check: none (judge only)
Report a problem with this finding
Typo Reference [14] reads "48 (1953) 503–30." with no comma after the year, unlike most entries (e.g. [10] "48 (1953), 486–502").
- p. 25 Reference list [14]
48 (1953) 503–30.
Judge's note: Read p.25 page image. [14] has no comma between "(1953)" and the pages. The same omission is in [17] "32 (1949) 119–34", and [5] and [9] omit the comma after the journal name. Punctuation inconsistency in the reference list only.
Second check: none (judge only)
Report a problem with this finding
Typo References [1], [2], [4], [5], [9], [20], [21] and [25] are listed but never cited in the text.
- p. 25 References [1], [2], [4], [5], [9]
[1] Bailey, W. G., and Haycocks, H. W.
- p. 26 References [20], [21], [25]
[20] Jablon, Seymour, "Testing of survival rates
Judge's note: The text layer of this scan has lost its digits, so I read every page image (pp.1-26) for bracketed citation numbers. Cited: [3], [6], [7], [8], [10]-[19], [22], [23], [24], [26], [27], [28]. No citation of [1], [2], [4], [5], [9], [20], [21] or [25] was found. Uncited background entries were common in papers of this period, so either form could be legitimate.
Second check: none (judge only)