Pith. sign in

REVIEW 3 major objections 5 minor 81 references

Estimating Distributions with Failure Rate Properties from Noisy Quantile Data

T0 review · 3 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows that estimating an increasing-failure-rate distribution from noisy quantile data reduces to a finite convex optimization problem, and its two-step algorithm provably solves the original infinite-dimensional maximum-likeliho

desk verdict Genuinely useful convex reformulation and solid numerics, but the main finite-sample bound rests on a separation event that is both uncontrolled and impossible near the upper endpoint, so the convergence claim in Theorem 2 is not proven. read the letter →

arxiv 2607.13212 v1 pith:TJTJ5ZJY submitted 2026-07-14 stat.AP

Timothy C.Y. Chan , Ningyuan Chen , Craig Fernandes , Muhammad Maaz This is my paper
classification stat.AP MSC 62G0562N0190C25
keywords increasingfailureratenoisyquantiledatacurrentstatusshape-constrainedestimationmaximumlikelihoodshape-preservinginterpolationfinite-sampleerrorboundsestimate-then-optimize
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tackles a common data situation: you do not see exact draws from a distribution, only binary yes/no answers at a few fixed thresholds (prices, inspection times, dose levels), and you want a full cdf that respects a shape constraint like increasing failure rate (IFR). The authors show that the natural maximum-likelihood problem is infinite-dimensional and nonconvex, but a change of variables—t(x) = ln(1−F(x))—turns it into a finite convex program over the knots plus a shape-preserving interpolation step. They prove the two-step algorithm returns an optimal solution to the original problem, and give finite-sample sup-norm bounds and convergence rates that translate into practical guidance on how to spend a data-collection budget. The framework extends to other failure-rate properties (IFRA, NBU, IGFR) and the case studies show better downstream pricing and maintenance decisions than benchmarks that ignore the shape constraint.

What carries the argument

The load-bearing object is the transformation t(x) = ln(1−F(x)), which maps the IFR property—non-decreasing hazard h = f/(1−F)—to concavity of t. The log-likelihood in t-space is concave in the knot values, so the infinite-dimensional nonconvex problem collapses to a finite convex program over the knot values τ_i = t(x_i). A second ingredient is shape-preserving interpolation (piecewise-linear or Schumaker quadratic spline) that extends the optimized knot values to a full concave, non-increasing t, hence an IFR cdf. The error analysis decomposes the sup-norm distance into a knot-level estimation error, controlled by Hoeffding's inequality under a separation condition, and an interpolation er

What would settle it

Run a simulation with a known IFR cdf that has very little mass near the lower end (so F0(l+ε) is below the assumed η), draw binomial samples at a few knots, and record how often the estimated Fhat at a knot falls within η of 0 or 1. Compare the empirical sup-norm error to the Proposition 1 bound conditional on the separation event; if the separation event fails often at moderate sample sizes, the unconditional 'with probability at least 1−δ' claim is not established.

Watch

Extended reading notes

Core claim

The central discovery is that the IFR constraint in F-space, expressed as log-concavity of the survival function, becomes concavity of t = ln(1−F). Since the binomial log-likelihood depends only on the knot values of t, solving the infinite-dimensional nonconvex problem is equivalent to solving a finite-dimensional convex program over the transformed knot values, then completing the curve by any non-increasing concave interpolant. Algorithm 1 does exactly this, and Theorem 1 shows its output is an optimal solution of the original problem. Theorem 2 bounds the sup-norm error between the estimated and true cdf on the full support by the t-space error on an interior interval plus a boundary ter

Load-bearing premise

The error bounds are conditioned on the estimated cdf Fhat—a random output of the algorithm—staying bounded away from 0 and 1 at every knot, an event whose probability is not bounded in the paper; if Fhat touches the boundary at any knot, the finite-sample guarantee does not apply.

Editorial extensions

If this is right

  • Practitioners can fit an IFR cdf from aggregated binary counts at a handful of thresholds using standard convex solvers, with a guarantee that they are solving the original nonconvex likelihood problem.
  • The error bound yields an offline data-collection rule: allocate samples equally across knots, place knots equidistantly, and let the number of knots grow slowly (roughly k ≍ (N/ln N)^(1/6)) with total budget, so add observations at existing knots before adding more.
  • The estimator converges in probability to the true cdf as the grid refines and per-knot sample sizes grow, with sup-norm error dominated by a O(1/√n) estimation term and a O(Δ²) or O(Δ³) interpolation term.
  • The same t-space reformulation extends to IFRA, NBU, and IGFR constraints, each expressible as linear inequalities at the knots, with grid-based interpolation for a full distributional estimate.
  • In the pricing and maintenance case studies, IFR-preserving fits lead to revenue and cost ratios close to the oracle, whereas benchmarks that ignore the shape constraint can yield multimodal objective curves and poor decisions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An implication the paper leaves implicit: because the boundary term max{F0(l+ε), 1−F0(xk)} controls the region beyond the largest knot, a practitioner who suspects the downstream optimum lies there (e.g., a high optimal price) could insert a few extra knots near that region to shrink the boundary term, at the cost of slightly worse knot-level estimation—a trade-off the paper does not quantify.
  • Beyond the paper's fixed-knot analysis, the error decomposition suggests an adaptive design: fit an initial coarse estimate, locate where F0 is steep or where the downstream objective is sensitive, then place new knots there in a second batch; this is a direct testable extension of the equal-allocation guidance.
  • A natural strengthening, not given in the paper, would be a two-stage bound: first show under mild conditions on F0 and the data that the separation event (Fhat bounded away from 0 and 1 at all knots) holds with high probability, then apply Proposition 1; without that, the stated finite-sample guarantee is conditional on an uncontrolled random event.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies estimation of a cumulative distribution function (cdf) satisfying increasing-failure-rate (IFR) and related shape constraints, from noisy quantile data: at each of finitely many knots, one observes binomial counts of draws below the knot. The authors reformulate the IFR-constrained maximum likelihood problem through the transformation t(x)=ln(1-F(x)), showing that the infinite-dimensional non-convex problem reduces to a finite-dimensional convex program over the knot values, followed by a shape-preserving interpolation step. They provide an optimality theorem for the algorithm, finite-sample sup-norm bounds in t- and F-space, convergence rates for piecewise-linear and Schumaker interpolation, and guidance for off-line data collection. The framework is extended to failure-rate average, new-better-than-used, and generalized failure-rate properties. Numerical experiments and two estimate-then-optimize case studies compare the method favorably against discretized benchmarks.

Significance. The paper addresses a genuinely useful and understudied problem: combining current-status/noisy-quantile data with failure-rate shape constraints. The change of variables t=ln(1-F) is elegant and appears to make the knot-level estimation problem convex for IFR and several related classes, which is a substantive technical contribution. The finite-sample bounds for fixed knots and the interpolation-error analysis are also valuable, and the numerical evidence is reasonably thorough. The provided code and reproduction of all experiments are strengths. However, the central asymptotic guarantee in Theorem 2 is not supported as stated, and Proposition 1 has a proof gap in its separation condition. If these theoretical issues are repaired, the paper would be a solid contribution to shape-constrained distribution estimation; in its current form the main convergence theorem overstates what is proven.

major comments (3)
  1. [Section 4.4, Theorem 2] The final convergence claim ('if the grid refines so that Δ→0, x_k→u, and n→∞, then ||F0−Fhat||_[l,u]→0 in probability') is not established. The theorem assumes a fixed ε>0 and a fixed η_ε>0 such that η_ε≤F0(x_i) and Fhat(x_i)≤1−η_ε for every knot in [l+ε,x_k]. As x_k→u, the last knot lies in this interval, so the condition (and the proof's use of F∈[η,1−η]) would require η_ε≤1−F0(x_k)→0. No fixed η_ε can satisfy this for all k, and the constant 1/[2η_ε^2(1−η_ε)] in the displayed bound is not uniform. Moreover, Fhat(x_k)≤1−η_ε is a random event whose probability is nowhere bounded. The terminal-region argument in the proof uses only monotonicity and therefore inherits this dependence. A correct proof would need either a sequence η_k→0 with a rate analysis and explicit control of P(Fhat(x_k)≤1−η_k), or a convergence statement restricted to a fixed compact interval plus a separate tail arg
  2. [Section 4.2, Proposition 1] The separation assumption is asymmetric: it states η≤F0(x_i) and Fhat(x_i)≤1−η for all i. In Step 6 of the proof, the translation to t-space uses the bound |ln(1−Fhat)−ln(1−F0)|≤(1/η)|Fhat−F0|, justified by 'F∈[η,1−η]'. But the stated assumption does not imply F0(x_i)≤1−η nor Fhat(x_i)≥η. Without a symmetric condition η≤F0(x_i),Fhat(x_i)≤1−η, the displayed t-space bound is not proven. This gap also propagates into Theorem 2, which uses Proposition 1 with the same asymmetric condition.
  3. [Section 3.1 and Remark 2] There is a disconnect between Algorithm 1 and its practical implementations. Algorithm 1, Step 2 requires a non-increasing concave interpolant satisfying t_hat(x)→−∞ as x→u. The piecewise-linear and Schumaker interpolants described in Section 3.1 cannot satisfy this over a finite interval; Remark 2 replaces the tail by a finite value −M, producing a jump at u so that the resulting Fhat is not differentiable (and not IFR in the strict sense used in (1)). Consequently, Theorem 1's claim that Algorithm 1 returns an optimal solution of (1) does not apply to the implemented versions. The authors should separate the theoretical algorithm (with a tail appended) from the practical approximation, or state Theorem 1 only for the theoretical interpolation operator.
minor comments (5)
  1. [Throughout] The notation for the smallest per-knot sample size n appears as both n and ar n; please make the notation consistent and define it before first use in Theorem 2.
  2. [Section 4.5] The data-collection balancing relation solves to approximately 3.1 and 4.35 knots for N=1,000 and 10,000. Since k is an integer, clarify that these are solutions of the continuous balancing equation, not literal integer recommendations.
  3. [Remark 2] The sentence beginning 'the left-limit induced by setting t_hat(u)=−M...' should state explicitly that this is a limit from the left and that Fhat(u)=1 creates a discontinuity at u, so the estimated cdf is not differentiable at that point.
  4. [Lemma 3 proof] The proof asserts that any discretely concave, non-positive knot vector admits a non-increasing concave interpolation on [l,x_k] and a terminal segment with t→−∞ on [x_k,u]. This is true, but the construction should be stated (e.g., a logarithmic tail), since the existence of the terminal segment is essential for the equivalence.
  5. [Table 1 caption] Typo: 'T able 1' should be 'Table 1'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimator's optimality and error bounds derive from standard MLE/Hoeffding and interpolation arguments; self-citations appear only in non-load-bearing literature review.

full rationale

The derivation chain is self-contained. Algorithm 1's optimality (Theorem 1) follows from Lemma 3, which proves equality of the objective values of (1) and (2) by a two-sided constructive argument: any feasible t gives knot values feasible for (2), and any feasible discrete knot vector is extended by a non-increasing concave interpolant and terminal segment. This is a mathematical equivalence, not a definitional shortcut. Proposition 1 bounds knot-level estimation error from the optimality inequality and Hoeffding/union bounds; the separation condition eta<=F0(xi), Fhat(xi)<=1-eta is a regularity assumption, not an input that is later relabeled as the output. Proposition 2 uses standard numerical-analysis interpolation-error bounds, and Theorem 2 combines the two components with a Lipschitz translation, then controls terminal regions by monotonicity and endpoint conditions. The numerical sections validate the rates against synthetic ground-truth distributions and compare with independent benchmarks (discretized-IFR, non-IFR, Weibull), so no fitted constant is fed back into the theory. Self-citations (Chen and Khademi 2024; Chen and Hu 2023) appear only in literature/application context and are not load-bearing. The known reparametrization t=ln(1-F) is standard and cited to Barlow and Proschan, not a renamed empirical pattern. A caveat: Proposition 1/Theorem 2's assumption on Fhat is not controlled, and the asymptotic path x_k->u makes a fixed eta_epsilon impossible; this is a correctness gap in the convergence claim, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard concentration and interpolation results plus four domain assumptions: the ground truth is IFR and smooth, data are nondegenerate at every knot, the estimator stays separated from 0/1 (the η-condition), and the interpolation operator preserves concavity/nonincreasingness. The free parameters are the separation constant η and the endpoint truncation constant M; neither is fitted to data, but the η-condition is not shown to hold with high probability.

free parameters (2)
  • separation constant η
    Assumed to exist in Proposition 1/Theorem 2 with η ≤ F0(xi), Fhat(xi) ≤ 1−η at all knots. It multiplies the error bound; no construction or probability control is given for the event involving the random Fhat.
  • endpoint truncation constant M
    Remark 2 replaces the exact right-endpoint condition t(u)=−∞ with t(u)=−M. The induced F-space error is at most e^{−M}; the paper does not specify M in the experiments.
assumptions (6)
  • domain assumption Ground-truth F0 is differentiable, supported on finite [l,u], and IFR (equivalently t0 = ln(1−F0) is concave).
    Section 2, equation (1e); the whole method optimizes over this class and the error bound is stated relative to F0.
  • domain assumption Nondegenerate data 0 < yi < ni at every knot.
    Lemma 2 uses this for existence and uniqueness of the convex program; the paper acknowledges degenerate cases occur with vanishing probability.
  • domain assumption Separation η exists for both F0 and Fhat at all knots.
    Proposition 1 bounds estimation error only when cdf values are bounded away from 0 and 1; since Fhat is random, this is a conditioning event.
  • domain assumption Smoothness t0 ∈ C² (piecewise-linear) or C³ (Schumaker).
    Proposition 2 and Corollary 2; the interpolation error bounds are meaningless if the true transformed cdf is less smooth.
  • standard math Schumaker interpolation and Fritsch–Butland derivative estimates preserve monotonicity and concavity with the stated accuracy.
    EC.3 proof of Proposition 2 uses shape preservation and a derivative-error bound from Fritsch and Butland (1984).
  • standard math Hoeffding and Pinsker inequalities hold as stated.
    Proposition 1 proof, Steps 2–5.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Estimating Distributions with Failure Rate Properties from Noisy Quantile Data." pith.science (2026). https://pith.science/paper/TJTJ5ZJY

@misc{pith2026260713212,
  author       = {Pith},
  title        = {Pith review of: Estimating Distributions with Failure Rate Properties from Noisy Quantile Data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJTJ5ZJY}},
  note         = {Machine review of arXiv:2607.13212}
}
read the original abstract

Estimating an unknown cumulative distribution function (cdf) from data, either as a statistical object of interest or as an input to a downstream optimization problem, is fundamental in operations. In practice, however, distribution estimation is often complicated by incomplete knowledge of the distribution's structure and limited, censored data. To address the first complication, we study distributions satisfying failure-rate shape constraints, especially increasing failure rate (IFR), rather than assuming a fully specified parametric family. To address the second, we consider noisy quantile data: at finitely many prespecified knots, each observation records only whether an independent sample lies below or above the knot. This combination arises naturally in pricing, reliability, and healthcare applications. We formulate the IFR-constrained maximum likelihood estimator and show that the original problem is infinite-dimensional and non-convex. We then develop a tractable two-step approach that solves a finite-dimensional convex optimization problem over transformed knot values and reconstructs a full cdf through shape-preserving interpolation. We establish finite-sample error bounds and convergence rates, yielding practical guidance for offline data collection. We also extend the framework to failure-rate-average, new-better-than-used, and generalized-failure-rate properties. Numerical experiments and case studies in revenue management and reliability demonstrate strong goodness-of-fit and improved downstream decision quality.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

81 extracted references · 1 linked inside Pith

  1. [1]

    1996 , publisher=

    Mathematical theory of reliability , author=. 1996 , publisher=

    1996

  2. [2]

    The Annals of Mathematical Statistics , pages=

    A note on tests for monotone failure rate based on incomplete data , author=. The Annals of Mathematical Statistics , pages=. 1969 , publisher=

    1969

  3. [3]

    SIAM Journal on Numerical Analysis , volume=

    On shape preserving quadratic spline interpolation , author=. SIAM Journal on Numerical Analysis , volume=. 1983 , publisher=

    1983

  4. [4]

    2016 , publisher=

    Nonlinear Programming , author=. 2016 , publisher=

    2016

  5. [5]

    1991 , publisher=

    Statistical Analysis of Reliability and Life-Testing Models: Theory and Methods , author=. 1991 , publisher=

    1991

  6. [6]

    Raja and Damaraju, C

    Rao, B. Raja and Damaraju, C. V. , title =. Biometrical Journal , volume =

  7. [7]

    Manufacturing & service operations management , volume=

    Selling to the newsvendor: An analysis of price-only contracts , author=. Manufacturing & service operations management , volume=. 2001 , publisher=

    2001

  8. [8]

    Operations Research , volume=

    A note on probability distributions with increasing generalized failure rates , author=. Operations Research , volume=. 2006 , publisher=

    2006

Show all 81 references
  1. [9]

    Mathematics of operations research , volume=

    Optimal auction design , author=. Mathematics of operations research , volume=. 1981 , publisher=

    1981

  2. [10]

    Economic Theory , volume=

    Regular type distributions in mechanism design and-concavity , author=. Economic Theory , volume=. 2013 , publisher=

    2013

  3. [11]

    Soviet Journal of Computer and Systems Sciences , volume=

    Quadratic optimization problems , author=. Soviet Journal of Computer and Systems Sciences , volume=

  4. [12]

    2011 , publisher=

    A first course on numerical methods , author=. 2011 , publisher=

    2011

  5. [13]

    SIAM journal on scientific and statistical computing , volume=

    A method for constructing local monotone piecewise cubic interpolants , author=. SIAM journal on scientific and statistical computing , volume=. 1984 , publisher=

    1984

  6. [14]

    2000 , publisher=

    Asymptotic statistics , author=. 2000 , publisher=

    2000

  7. [15]

    Journal of the American Statistical Association , volume=

    Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions , author=. Journal of the American Statistical Association , volume=. 1987 , publisher=

    1987

  8. [16]

    2009 , publisher=

    Variational analysis , author=. 2009 , publisher=

    2009

  9. [17]

    2013 , publisher =

    Maintenance, Replacement, and Reliability: Theory and Applications , author =. 2013 , publisher =

    2013

  10. [18]

    Smith, W. L. , title =. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences , volume =

  11. [19]

    Scandinavian Actuarial Journal , volume =

    Ulf Grenander , title =. Scandinavian Actuarial Journal , volume =. 1956 , doi =

    1956

  12. [20]

    Bernoulli , volume =

    Lutz Dümbgen and Kaspar Rufibach , title =. Bernoulli , volume =. 2009 , doi =

    2009

  13. [21]

    Marshall and Frank Proschan , title =

    Albert W. Marshall and Frank Proschan , title =. The Annals of Mathematical Statistics , volume =

  14. [22]

    W. J. Padgett and L. J. Wei , title =. Biometrika , volume =. 1980 , doi =

    1980

  15. [23]

    Economic Theory , volume =

    Mark Bagnoli and Ted Bergstrom , title =. Economic Theory , volume =. 2005 , doi =

    2005

  16. [24]

    E. L. Kaplan and Paul Meier , title =. Journal of the American Statistical Association , volume =

  17. [25]

    Annals of Statistics , volume =

    Odd Aalen , title =. Annals of Statistics , volume =. 1978 , doi =

    1978

  18. [26]

    Journal of Quality Technology , volume =

    Wayne Nelson , title =. Journal of Quality Technology , volume =

  19. [27]

    Technometrics , volume =

    Wayne Nelson , title =. Technometrics , volume =

  20. [28]

    B. L. S. Prakasa Rao , title =. The Annals of Mathematical Statistics , volume =

  21. [29]

    R. E. Barlow and D. J. Bartholomew and J. M. Bremner and H. D. Brunk , title =. 1972 , address =

    1972

  22. [30]

    Journal of Statistical Software , volume =

    de Leeuw, Jan and Hornik, Kurt and Mair, Patrick , title =. Journal of Statistical Software , volume =. 2009 , doi =

    2009

  23. [31]

    Statistics in Biopharmaceutical Research , volume=

    Centered isotonic regression: point and interval estimation for dose--response studies , author=. Statistics in Biopharmaceutical Research , volume=

  24. [32]

    Operations Research , volume =

    Felipe Caro and Jérémie Gallien , title =. Operations Research , volume =. 2012 , doi =

    2012

  25. [33]

    Manufacturing & Service Operations Management , volume =

    Kris Johnson Ferreira and Bin Hong Alex Lee and David Simchi-Levi , title =. Manufacturing & Service Operations Management , volume =. 2015 , doi =

    2015

  26. [34]

    Elmachtoub and Yunjie Sun , title =

    Omar Besbes and Adam N. Elmachtoub and Yunjie Sun , title =. INFORMS Journal on Applied Analytics , volume =. 2020 , doi =

    2020

  27. [35]

    Manufacturing & Service Operations Management , volume =

    Pol Boada-Collado and Victor Martínez-de-Albéniz , title =. Manufacturing & Service Operations Management , volume =. 2019 , doi =

    2019

  28. [36]

    Arslan and Robert F

    Hayri A. Arslan and Robert F. Easley and Ruxian Wang and Övünç Yılmaz , title =. Manufacturing & Service Operations Management , volume =. 2021 , doi =

    2021

  29. [37]

    Service Science , volume =

    Ningyuan Chen and Ming Hu , title =. Service Science , volume =

  30. [38]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume =

    Bryan Wilder and Bistra Dilkina and Milind Tambe , title =. Proceedings of the AAAI Conference on Artificial Intelligence , volume =. 2019 , doi =

    2019

  31. [39]

    Proceedings of the 39th International Conference on Machine Learning , series =

    Jayanta Mandi and Vı́ctor Bucarey and Maxime Mulamba and Tias Guns , title =. Proceedings of the 39th International Conference on Machine Learning , series =

  32. [40]

    Operations Research , volume =

    Gah-Yi Ban and Cynthia Rudin , title =. Operations Research , volume =

  33. [41]

    Elmachtoub and Paul Grigas , title =

    Adam N. Elmachtoub and Paul Grigas , title =. Management Science , volume =. 2021 , doi =

    2021

  34. [42]

    Management Science , volume =

    Dimitris Bertsimas and Nathan Kallus , title =. Management Science , volume =. 2019 , doi =

    2019

  35. [43]

    Proceedings of the Sixteenth ACM Conference on Economics and Computation (EC '15) , pages =

    Hu Fu and Nicole Immorlica and Brendan Lucier and Philipp Strack , title =. Proceedings of the Sixteenth ACM Conference on Economics and Computation (EC '15) , pages =. 2015 , doi =

    2015

  36. [44]

    Proceedings of the Sixteenth ACM Conference on Economics and Computation (EC '15) , pages =

    Zhiyi Huang and Yishay Mansour and Tim Roughgarden , title =. Proceedings of the Sixteenth ACM Conference on Economics and Computation (EC '15) , pages =. 2015 , doi =

    2015

  37. [45]

    Proceedings of the 21st ACM Conference on Economics and Computation (EC '20) , pages =

    Constantinos Daskalakis and Manolis Zampetakis , title =. Proceedings of the 21st ACM Conference on Economics and Computation (EC '20) , pages =. 2020 , doi =

    2020

  38. [46]

    Operations Research , volume =

    Amine Allouah and Achraf Bahamou and Omar Besbes , title =. Operations Research , volume =. 2022 , doi =

    2022

  39. [47]

    Gonczarowski and Yishay Mansour and Shay Moran , title =

    Moshe Babaioff and Yannai A. Gonczarowski and Yishay Mansour and Shay Moran , title =. Proceedings of the 2018 ACM Conference on Economics and Computation (EC '18) , pages =. 2018 , publisher =

    2018

  40. [48]

    2024 , eprint =

    Achraf Bahamou and Omar Besbes and Omar Mouchtaki , title =. 2024 , eprint =

    2024

  41. [49]

    Management Science , volume =

    Amine Allouah and Achraf Bahamou and Omar Besbes , title =. Management Science , volume =. 2023 , doi =

    2023

  42. [50]

    2024 , note =

    Ali Daei Naby and Setareh Farajollahzadeh and Ming Hu , title =. 2024 , note =

    2024

  43. [51]

    Journal of the American Statistical Association , volume =

    Wassily Hoeffding , title =. Journal of the American Statistical Association , volume =

  44. [52]

    Sun, Jianguo , title =

  45. [53]

    Turnbull , journal =

    Bruce W. Turnbull , journal =. The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data , volume =

  46. [54]

    Journal of the Royal Statistical Society Series A: Statistics in Society , volume =

    Keiding, Niels , title =. Journal of the Royal Statistical Society Series A: Statistics in Society , volume =

  47. [55]

    , title =

    Groeneboom, Piet and Wellner, Jon A. , title =. Information Bounds and Nonparametric Maximum Likelihood Estimation , pages =

  48. [56]

    Groeneboom, Piet and Jongbloed, Geurt , title =

  49. [57]

    , title =

    Huang, Jian and Wellner, Jon A. , title =. Proceedings of the First Seattle Symposium in Biostatistics , editor =

  50. [58]

    The annals of mathematical statistics , volume =

    An empirical distribution function for sampling with incomplete information , author=. The annals of mathematical statistics , volume =

  51. [59]

    Consistency of Concave Regression with an Application to Current-Status Data , journal =

    D. Consistency of Concave Regression with an Application to Current-Status Data , journal =

  52. [60]

    Estimating a Unimodal Distribution from Interval-Censored Data , journal =

    D. Estimating a Unimodal Distribution from Interval-Censored Data , journal =

  53. [61]

    Maximum-Likelihood Estimation of a Log-Concave Density Based on Censored Data , journal =

    D. Maximum-Likelihood Estimation of a Log-Concave Density Based on Censored Data , journal =

  54. [62]

    Statistics and Computing , volume =

    Anderson-Bergman, Clifford and Yu, Yun , title =. Statistics and Computing , volume =

  55. [63]

    arXiv preprint arXiv:2411.19878 , year=

    Nonparametric Estimation for a Log-concave Distribution Function with Interval-censored Data , author=. arXiv preprint arXiv:2411.19878 , year=

    arXiv

  56. [64]

    2019 , publisher =

    Revenue Management and Pricing Analytics , author =. 2019 , publisher =

    2019

  57. [65]

    2009 , publisher =

    Introduction to Nonparametric Estimation , author =. 2009 , publisher =

    2009

  58. [66]

    2002 , publisher =

    Statistical Inference , author =. 2002 , publisher =

    2002

  59. [67]

    Technometrics , volume =

    Theoretical Explanation of Observed Decreasing Failure Rate , author =. Technometrics , volume =

  60. [68]

    AT&T Technical Journal , volume =

    An Infant Mortality and Long-Term Failure Rate Model for Electronic Equipment , author =. AT&T Technical Journal , volume =

  61. [69]

    British Journal of Cancer , volume =

    Survival Analysis Part I: Basic Concepts and First Analyses , author =. British Journal of Cancer , volume =. 2003 , doi =

    2003

  62. [70]

    Journal of Machine Learning Research , year =

    Steven Diamond and Stephen Boyd , title =. Journal of Machine Learning Research , year =

  63. [71]

    Operations Research , volume =

    Relationships Among Three Assumptions in Revenue Management , author =. Operations Research , volume =

  64. [72]

    Journal of Control and Decision , year =

    Agrawal, Akshay and Verschueren, Robin and Diamond, Steven and Boyd, Stephen , title =. Journal of Control and Decision , year =

  65. [73]

    1998 , publisher=

    Numerical methods in economics , author=. 1998 , publisher=

    1998

  66. [74]

    PeerJ Computer Science , issn =

    SymPy: symbolic computing in Python , author =. PeerJ Computer Science , issn =

  67. [75]

    INFORMS Journal on Computing , year =

    Bertsimas, Dimitris and Mundru, Nishanth , title =. INFORMS Journal on Computing , year =

  68. [76]

    Statistical Science , year =

    Guntuboyina, Adityanand and Sen, Bodhisattva , title =. Statistical Science , year =

  69. [77]

    , title =

    Lim, Eunji and Glynn, Peter W. , title =. Operations Research , year =

  70. [78]

    Mathematical Programming Computation , year =

    Lin, Meixia and Sun, Defeng and Toh, Kim-Chuan , title =. Mathematical Programming Computation , year =

  71. [79]

    Manufacturing & Service Operations Management , year =

    Chen, Ningyuan and Khademi, Amin , title =. Manufacturing & Service Operations Management , year =

  72. [80]

    , title =

    Manski, Charles F. , title =. Epidemiology , year =

  73. [81]

    Biometrics , volume=

    An Increasing Failure Rate Approach to Low-Dose Extrapolation , author=. Biometrics , volume=

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.