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For generalized progressive hybrid censored life tests, the paper claims that precision improves while cost worsens as the censoring time or the guaranteed number of failures increases—so the best design under a budget is found by solving o

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For generalized progressive hybrid censoring experiments, the paper claims A-optimal designs can be found by pushing censoring time to its budget boundary and searching the removal vector with a heuristic, and it derives a Shannon-entropy criterion.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection The A-optimality/cost framework for GPHCS-I is a solid, useful extension, but the entropy theorem at the paper's core is unproved and almost certainly false for the Weibull models used. the 4 major comments →

arxiv 2607.14926 v1 pith:DF6FGQ2U submitted 2026-07-16 stat.AP cs.ITmath.IT

Optimal Design for Generalized Progressive Hybrid Censored Data via Constrained, Unconstrained, Compound, and Minimax Optimization

Rathin Das , Tanmay Sen , Deepak Prajapati This is my paper
classification stat.AP cs.ITmath.IT MSC 62K0562N0562N01
keywords A-optimal designEntropy optimizationMulti-objective optimizationMinimax optimizationVariable neighbourhood searchWeibull distributionCompound designGeneralized progressive hybrid censoring
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

This paper tries to establish a practical recipe for designing a Type-I generalized progressive hybrid censoring life test—choosing the inspection time T, the guaranteed minimum number of failures l, and the progressive removal vector R—by proving that the design problem has far less structure to search over than a naive three-variable optimization would suggest. Its central structural claim is a clean conflict: the A-optimality criterion (the trace of the inverse Fisher information, i.e., total estimator variance) falls as T or l grows, while expected total cost rises as T or l grows. Consequently, under a fixed budget the best T for any candidate (R,l) sits at the budget boundary TC = CB, and the search reduces to a one-dimensional sweep over l for each R, with a Variable Neighborhood Search handling R. The paper also claims a closed-form Shannon entropy for GPHCS-I data, decomposed as H_{1..l} + H_{X_m∧T} − H_{X_l∧T}, and uses it as a complementary criterion that ranks designs by observed-data uncertainty rather than estimation precision. A sympathetic reader should care because the result turns a computationally heavy combinatorial design problem into an efficient algorithm and supplies an information-theoretic check that often disagrees with A-optimality.

Core claim

The central discovery, stated on the paper's own terms, is that the Fisher information and the Shannon entropy of GPHCS-I data both split additively: I_{X_l∨(X_m∧T)} = I_{1..l} + I_{X_m∧T} − I_{X_l∧T} and H_{X_l∨(X_m∧T)} = H_{1..l} + H_{X_m∧T} − H_{X_l∧T}. From the information identity the paper proves that the A-criterion decreases in both T and l; from its cost model, expected total cost increases in both. These opposing monotonicities turn the constrained design problem into a one-dimensional root-finding problem for T at the budget boundary plus a one-dimensional search in l and a heuristic search in R. The paper then shows numerically that GPHCS-I designs beat conventional progressive h

What carries the argument

The load-bearing object is the stopping time τ = max{X_l:m:n, min(X_m:m:n, T)} together with the additivity identity for Fisher information and its Shannon-entropy counterpart. The identity decomposes a GPHCS-I sample into the first l progressively censored order statistics plus the tail of failures truncated at T, identified as a difference of two PHCS-I entropies. The monotonicity theorems (A-criterion decreases, cost increases, in T and l) turn the feasible region into a thin boundary, and the Variable Neighborhood Search—a heuristic that explores progressively larger neighborhoods of a current removal vector—handles the remaining combinatorial piece R.

Load-bearing premise

The load-bearing premise is the identity H_{X_l∨(X_m∧T)} = H_{1..l} + H_{X_m∧T} − H_{X_l∧T}: the proof identifies the conditional entropy of the tail after the l-th failure with a difference of two PHCS-I entropies, and for non-exponential lifetimes that conditional integration is asserted rather than shown; if the equality fails, all entropy-optimal designs in the numerical tables lose their basis.

What would settle it

Pick a Weibull(2,1) model with fixed n, m, R, l and several T values; compute the left side H_{X_l∨(X_m∧T)} by direct numerical integration of the GPHCS-I joint density (or by Monte Carlo histogram entropy) and compare with the right side H_{1..l} + H_{X_m∧T} − H_{X_l∧T}. Any systematic mismatch beyond numerical error refutes Theorem 3.1; a match supports it. A secondary calculation: verify the monotonicity claims by computing the A-criterion and expected cost at increasing T and l.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • For any fixed removal vector R and guaranteed-failure count l, the optimal censoring time under an active budget is obtained by solving TC = CB, not by grid search.
  • The optimal guaranteed-failure count for each R can be found by sweeping l upward until the budget is infeasible, because cost increases with l; this prunes the search space.
  • GPHCS-I with a positive guaranteed minimum number of failures yields A-optimality and entropy values at least as good as PHCS-I under the same budget, with the largest gains under tight budgets and increasing hazard rates (per the paper's Weibull numerical studies).
  • The minimax multi-objective formulation balances precision and cost without user-selected weights and extends to a third objective (entropy), while compound designs need a weight sweep and trade-off plots.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference — the same conflicting-monotonicity argument should transfer to other hybrid censoring schemes (Type-II generalized progressive hybrid, or hybrid rules with different termination conditions) whenever the information measure is additive and the cost is increasing in the amount of observation; testing this would clarify how general the boundary-solution phenomenon is.
  • Editorial inference — the entropy identity's dependence on the Markov property of order statistics suggests a direct simulation check for non-exponential lifetimes; if the identity fails for small n or heavy censoring, the A-optimality framework survives but the entropy tables would need re-derivation.
  • Editorial inference — because entropy and A-optimality pick different designs, the gap between the two criteria could be used as a diagnostic for whether a chosen design is information-rich versus parameter-precise, or as a third objective in a minimax search that explicitly trades all three.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The manuscript develops optimal design methodology for Type-I generalized progressive hybrid censoring (GPHCS-I) life tests with decision variables (R, T, l). It proposes a cost-constrained A-optimal design framework, claiming that the A-optimality criterion decreases with T and l while total expected cost increases, so the budget-constrained problem reduces to solving TC = C_B for T and searching over the removal vector R with a VNS heuristic. It also derives a Shannon entropy expression for GPHCS-I data, computes entropy-optimal designs, and formulates compound and minimax multi-objective criteria. Numerical studies are reported for Weibull lifetimes and for a real insulating-fluid data set.

Significance. If the results were correct, the paper would offer a practically useful way to reduce the computational burden of designing GPHCS-I experiments by exploiting monotonicity, and it would provide an information-theoretic criterion alongside A-optimality. The structural monotonicity idea, the budget-boundary reduction, and the VNS framework are attractive, and the numerical study is extensive. However, the central entropy identity, the proof of the A-optimality monotonicity theorem, and the minimax lemmas all contain serious errors that affect the reported optimal designs. These issues are load-bearing rather than cosmetic.

major comments (4)
  1. [§3.2, Theorem 3.1 and Eq. (5)] The claimed entropy identity H_{X_l∨(X_m∧T)} = H_{1..l} + H_{X_m∧T} - H_{X_l∧T} is not established and is generally false. In the proof, the conditional entropy H_{j|j-1} is replaced by H_{Z_{1:N_{j-1}}∧T}, a fixed entropy of a minimum of untruncated lifetimes. But Y_j given Y_{j-1}=y is governed by the residual distribution (F(t)-F(y))/(1-F(y)) with truncation point T-y, and it is degenerate if Y_{j-1} is already censored. Its entropy depends on y and must be integrated against the law of Y_{j-1}; the proof neither performs this integration nor justifies the substitution. For the exponential/Weibull models used later, a direct calculation in the m=2, l=1 case shows the two sides differ. In addition, Eq. (5) defines H_{X_i:m:n∧T} with the inner sum running to m for every i, so H_{X_m∧T}-H_{X_l∧T} would be zero, contradicting the proof's cancellation. Since Tables 2–5 and 8 and the 'All-t
  2. [§4, Theorem 4.5] The proof does not prove the stated monotonicity claims. Part (I) compares l and l+1 while the theorem statement says T; part (II) compares T1 and T2 while the statement says l. In part (II), I(ζ1)-I(ζ2) is written as ∫_{T2}^{T1} ..., which has the wrong orientation when T2>T1, and the same integral is then called I(ζ2)-I(ζ1) and asserted non-negative definite. In part (I), the correct difference in Fisher information when l increases is ∫_T^∞ f_{(l+1):m:n}(x) dx, not ∫_0^T. The monotonicity direction may be repairable, but the proof as printed is invalid, and Algorithm 1's boundary argument depends on this theorem.
  3. [§5.2, Lemma 5.2 and Theorem 5.3] The sign of g(T)=φ1(R,T,l)-φ2(R,T,l) is reversed. With φ1 the A-optimality criterion (decreasing in T by Theorem 4.5) and φ2 the total cost (increasing in T by Theorem 4.4), g is strictly decreasing, not increasing. Consequently, Theorem 5.3 parts (2) and (3) state the wrong boundary cases: if g<0 everywhere, Φ=φ2 is increasing, so T*→0, not ∞; if g>0 everywhere, Φ=φ1 is decreasing, so T*→∞, not 0. Algorithm 4 uses these boundary rules, so the minimax designs in Tables 6–7 and Table 8 are not justified by the stated theory.
  4. [§4, Proposition 4.3] The simplification E[D] = l + Σ_{d=l}^m F_d(T) is algebraically wrong. Telescoping the preceding expression gives E[D] = l + Σ_{d=l+1}^m F_d(T); the F_l term is spurious. Since E[D] enters the total cost and Algorithms 1 and 2 solve TC(R,T,l)=C_B to obtain T*, the reported constrained T* values and feasibility checks inherit this error. The monotonicity conclusion still holds, but the numerical cost calculations need to be redone.
minor comments (5)
  1. [§5.1] The notation is self-contradictory: the text says φ1 is A-optimality and φ2 is total cost, but then writes φ1(ζ)=TC(ζ) and φ2(ζ)=φ(ζ).
  2. [§4, Algorithms 1–2] The pseudocode says 'foreach feasible progressive censoring vector R' and then 'apply the VNS algorithm'; this does not define a concrete search. The VNS neighborhood structure and parameters are not specified, so the claim that R* is a global optimum is not verifiable.
  3. [§6.2, Tables 2–5] The captions and headers contain inconsistencies: 'PHCS-I-I' appears in Table 3, and the RRS/REL definitions use φ1/φ while the tables report φ and φ2.
  4. [§6.3, Tables 6–7] The RRS1/RRS2 columns appear to report normalized efficiencies rather than the percentage deviations defined in the text; the definitions and reported values do not match.
  5. [Algorithm 2] The line 'Determine l*(R) = arg min_l φE(R,T*,.' is truncated and incomplete.

Circularity Check

0 steps flagged

No significant circularity: the optimization criterion uses a self-cited but external Fisher information input, and the entropy theorem's gap is an unsupported step rather than a definitional reduction.

full rationale

The paper's claimed derivation chain is not circular. The A-optimality criterion is built on the Fisher information matrix cited from Sen et al. (2018). This is a self-citation, but it is an external published input: the paper's monotonicity results, budget-constrained boundary search, and VNS algorithm operate on that matrix rather than defining the target design in terms of the quantities they seek. No fitted constants are used to define any criterion; Weibull parameters and cost coefficients are fixed before optimization, and the reported designs are not reverse-engineered from the objective values. The cost-constrained reduction to TC(ζ)=CB follows from monotonicity of expected duration, expected failures, and cost, with the A-optimality monotonicity an algebraic consequence of the information-matrix decomposition. The entropy theorem in Section 3.2 is assembled from external formulas (Balakrishnan et al. 2007; Almohaimeed 2017). Its proof contains the asserted equality H_{j|j-1}=H_{Z_{1:N_{j-1}}∧T}, which is not justified and appears to fail for general Weibull lifetimes; this is a serious correctness gap that threatens the entropy-optimal tables, but it is not a circular reduction because the theorem's conclusion is not made true by definition of a fitted quantity or by an identity that is its own input. The proof merely asserts a stronger, unproved identity. Therefore no load-bearing step in the paper reduces by construction to its own inputs, and the appropriate finding is no significant circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central mathematical work rests on standard PSD-monotonicity plus domain assumptions about the lifetime distribution, the cost model, and the heuristic's global convergence. The paper introduces no new physical entities. The main ad hoc choices are the Tmin/Tmax truncation and the unverified VNS optimality.

free parameters (5)
  • Cost coefficients C0, Cτ, CD = C0=50; Cτ=25/50/75; CD=3/7/10 in numerical studies
    Hand-chosen economic inputs; all constrained and compromise designs are functions of these values.
  • Budget CB = 100, 125, 150 in simulations; 65 in data example
    Chosen by hand to define the constraint; the optimal T* is sensitive to CB.
  • Weibull parameters α, η = α=0.5,1,2, η=1; data example α=0.974, η=0.108 from Chakraborty et al. (2025)
    Plugged in as known values; all reported optimal designs are local to these parameters.
  • Tmin and Tmax = 0.01th and 0.99th quantiles of the assumed lifetime distribution
    Computational replacements for T=0 and T=∞; the unconstrained A-optimal and cost-optimal designs are reported at these bounds.
  • VNS neighborhood parameters = not specified
    Variable Neighborhood Search requires concrete neighborhood structures and local-search settings; without them the VNS search is not reproducible and global optimality is not certified.
axioms (6)
  • standard math If A and B are positive semidefinite with A ≤ B, then B^{-1} ≤ A^{-1} in Loewner order and tr(A^{-1}) ≥ tr(B^{-1}).
    Invoked in the proof of Theorem 4.5 to convert information-matrix monotonicity into A-optimality monotonicity.
  • domain assumption The lifetime distribution F(·|θ) is known and correctly specified, and all design criteria are evaluated at a single θ.
    All reported designs are local designs; if the true parameters differ, the selected schemes need not be optimal for the true model.
  • domain assumption Progressive removals R are preassigned and independent of the observed lifetimes.
    This is the standard progressive censoring model; adaptive or data-dependent removals would change the likelihood and the design problem.
  • domain assumption The total cost is exactly TC = C0 + Cτ E[τ] + CD E[D] with known coefficients.
    The constrained search is driven by this cost model; any different cost structure would change the feasible boundary and the reported optimal designs.
  • ad hoc to paper T=0 and T=∞ can be replaced by finite Tmin and Tmax, taken as the 0.01th and 0.99th quantiles.
    Theoretical optima at 0 and ∞ are not computable; all 'optimal' statements are relative to this truncation.
  • ad hoc to paper The VNS heuristic returns the global optimum over the removal-vector space.
    No optimality certificate, exhaustive check, or explicit VNS parameter settings are provided, yet outputs are labeled optimal.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Optimal Design for Generalized Progressive Hybrid Censored Data via Constrained, Unconstrained, Compound, and Minimax Optimization." pith.science (2026). https://pith.science/paper/DF6FGQ2U

@misc{pith2026260714926,
  author       = {Pith},
  title        = {Pith review of: Optimal Design for Generalized Progressive Hybrid Censored Data via Constrained, Unconstrained, Compound, and Minimax Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DF6FGQ2U}},
  note         = {Machine review of arXiv:2607.14926}
}
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read the original abstract

This paper studies the optimal design of Type-I generalized progressive hybrid censoring schemes for life-testing experiments. The design problem involves simultaneously determining the inspection time, the guaranteed number of failures, and the progressive censoring scheme. First we develop a cost-constrained optimization framework for determining the optimal censoring scheme. Structural properties of the A-optimality criterion and the experimental cost with respect to the inspection time and the guaranteed number of failures are established. It reveals that they are conflicting behaviors which enables to develop an efficient search algorithm that substantially reduces the computational burden. Building on these theoretical results, a multi-objective optimization model is proposed to simultaneously minimize A-optimality criterion and the experimental cost. A Variable Neighborhood Search (VNS) algorithm is proposed to efficiently determine the optimal progressive removal vector by exploring the feasible design space while avoiding exhaustive enumeration. The resulting compromise designs simultaneously improve estimation precision and reduce experimental cost. In addition, the Shannon differential entropy of the observed lifetime distribution is derived and employed as a complementary information-theoretic measure for evaluating the selected censoring schemes. Numerical studies show that entropy-optimal designs generally differ from A-optimal designs, indicating that Shannon entropy characterizes uncertainty in the observed data rather than estimation precision. The proposed methodology provides an efficient computational framework for optimal life-test design and offers a foundation for future multi-objective optimization incorporating statistical efficiency, experimental cost, and information-theoretic uncertainty.

Figures

Figures reproduced from arXiv: 2607.14926 by Deepak Prajapati, Rathin Das, Tanmay Sen.

Figure 1
Figure 1. Figure 1: Compound objective function plots for the Weibull distribution with [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Compound objective function plots for the Weibull distribution with [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Compound objective function plots for the Weibull distribution with [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Compound objective function plots corresponding to the sensitivity analysis reported in Table 7. [PITH_FULL_IMAGE:figures/full_fig_p028_4.png] view at source ↗

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Reference graph

Works this paper leans on

77 extracted references

  1. [1]

    2015 , publisher=

    Xu, Ancha and Tang, Yincai , journal=. 2015 , publisher=

    2015

  2. [2]

    Linhart, Heinz and Zucchini, Walter , year=

  3. [3]

    2023 , publisher=

    Chakraborty, Siddhartha and Bhattacharya, Ritwik and Pradhan, Biswabrata , journal=. 2023 , publisher=

    2023

  4. [4]

    2015 , publisher=

    Cho, Youngseuk and Sun, Hokeun and Lee, Kyeongjun , journal=. 2015 , publisher=

    2015

  5. [5]

    2001 , publisher=

    Gupta, Rameshwar D and Kundu, Debasis , journal=. 2001 , publisher=

    2001

  6. [6]

    D. T. Patel and M. N. Patel. International Journal of Statistics and Economics. 2017

    2017

  7. [7]

    2018 , publisher=

    Sen, Tanmay and Pradhan, Biswabrata and Tripathi, Yogesh Mani and Bhattacharya, Ritwik , journal=. 2018 , publisher=

    2018

  8. [8]

    2021 , publisher=

    Singh, Devendra Pratap and Lodhi, Chandrakant and Tripathi, Yogesh Mani and Wang, Liang , journal=. 2021 , publisher=

    2021

  9. [9]

    2022 , publisher=

    Elshahhat, Ahmed and Abu El Azm, Wael S , journal=. 2022 , publisher=

    2022

  10. [10]

    2024 , publisher=

    Singh, Kundan and Kumar Mahto, Amulya and Mani Tripathi, Yogesh , journal=. 2024 , publisher=

    2024

  11. [11]

    2024 , publisher=

    Xin, Ying and Zhou, Bingchang and Tang, Yaning and Zhang, You , journal=. 2024 , publisher=

    2024

  12. [12]

    2024 , publisher=

    Hassan, Amal S and Mousa, Rana M and Abu-Moussa, Mahmoud H , journal=. 2024 , publisher=

    2024

  13. [13]

    T. R. Tsai. International Journal of Reliability, Quality and Safety Engineering. 2008

    2008

  14. [14]

    Lee, Kyeong-Jun and Lee, Jae-Ik and Park, Chan-Keun , journal=

  15. [15]

    2007 , publisher=

    Manna, DK and Pal, Surajit and Sinha, Sagnik , journal=. 2007 , publisher=

    2007

  16. [16]

    2006 , publisher=

    Manna, DK and Pal, Surajit and Sinha, Sagnik , journal=. 2006 , publisher=

    2006

  17. [17]

    2008 , publisher=

    Manna, DK and Pal, Surajit and Sinha, Sagnik , journal=. 2008 , publisher=

    2008

  18. [18]

    J. B. Chakrabarty and S. Chowdhury and S. Roy. International Journal of Quality & Reliability Management. 2020

    2020

  19. [19]

    Y. I. Kwon. International Journal of Quality & Reliability Management. 1996

    1996

  20. [20]

    Aslam and R

    M. Aslam and R. Yousaf and S. Ali. Proceedings of the National Academy of Sciences, India Section A: Physical Sciences. 2020

    2020

  21. [21]

    Salem and Z

    M. Salem and Z. Amin and M. Ismail. Sankhya B. 2020

    2020

  22. [22]

    N. L. Johnson and S. Kotz and N. Balakrishnan. 1994

    1994

  23. [23]

    Reliability Engineering & System Safety , volume=

    Warranty forecasting from incomplete two-dimensional warranty data , author=. Reliability Engineering & System Safety , volume=. 2014 , publisher=

    2014

  24. [24]

    Quality Engineering , volume=

    Field reliability modeling based on two-dimensional warranty data with censoring times , author=. Quality Engineering , volume=. 2017 , publisher=

    2017

  25. [25]

    Guti \'e rrez-Pulido and V

    H. Guti \'e rrez-Pulido and V. Aguirre-Torres and J. A. Christen. A practical method for obtaining prior distributions in reliability. IEEE Transactions on Reliability. 2005

    2005

  26. [26]

    S. J. Wu and S. R. Huang. IEEE Transactions on Reliability. 2010

    2010

  27. [27]

    Balakrishnan and D

    N. Balakrishnan and D. Kundu. Hybrid censoring: models, inferential results and applications (with discussions). Computational Statistics & Data Analysis. 2013

    2013

  28. [28]

    A. C. Cohen. Progressively censored samples in life testing. Technometrics. 1963

    1963

  29. [29]

    A. C. Cohen. Technometrics. 1966

    1966

  30. [30]

    Chen and G

    S. Chen and G. K. Bhattacharyya. Communications in Statistics - Theory and Methods. 1988

    1988

  31. [31]

    Childs and B

    A. Childs and B. Chandrasekar and N. Balakrishnan and D. Kundu. Annals of the Institute of Statistical Mathematics. 2003

    2003

  32. [32]

    1956 , publisher=

    Estimation of the parameters of a population from a multi-censored sample , author=. 1956 , publisher=

    1956

  33. [33]

    2012 , publisher=

    Theory of optimal designs , author=. 2012 , publisher=

    2012

  34. [34]

    Statistical Methodology , volume=

    Exact likelihood inference for an exponential parameter under generalized progressive hybrid censoring scheme , author=. Statistical Methodology , volume=. 2015 , publisher=

    2015

  35. [35]

    2007 , publisher=

    Balakrishnan, Narayanaswamy and Rad, Arezou Habibi and Arghami, Naser Reza , journal=. 2007 , publisher=

    2007

  36. [36]

    2020 , publisher=

    Sen, Tanmay and Bhattacharya, Ritwik and Tripathi, Yogesh Mani and Pradhan, Biswabrata , journal=. 2020 , publisher=

    2020

  37. [37]

    Technometrics , volume=

    Interval estimation of parameters of life from progressively censored data , author=. Technometrics , volume=. 1994 , publisher=

    1994

  38. [38]

    B. Epstein. Annals of Mathematical Statistics. 1954

    1954

  39. [39]

    Kamps and E

    U. Kamps and E. Cramer. Statistics. 2001

    2001

  40. [40]

    D. Kundu. Journal of Statistical Planning and Inference. 2007

    2007

  41. [41]

    M. P. Kaminskiy and V. V. Krivtsov. IEEE Transactions on Reliability. 2005

    2005

  42. [42]

    Kundu and A

    D. Kundu and A. Joarder. Computational Statistics & Data Analysis. 2006

    2006

  43. [43]

    D. Kundu. Technometrics. 2008

    2008

  44. [44]

    Y. Lam. The Annals of Statistics. 1994

    1994

  45. [45]

    G. J. Lieberman and G. J. Resnikoff. Journal of the American Statistical Associstion. 1955

    1955

  46. [46]

    Y. P. Lin and T. Liang and W. T. Huang. Annals of the Institute of Statistical Mathematics. 2002

    2002

  47. [47]

    Park and N

    S. Park and N. Balakrishnan. Statistics and Probability Letters. 2009

    2009

  48. [48]

    Park and N

    S. Park and N. Balakrishnan and G. Zheng. Statistics and Probability Letters. 2008

    2008

  49. [49]

    Park and N

    S. Park and N. Balakrishnan and S. W. Kim. Statistics. 2011

    2011

  50. [50]

    Clyde and P

    M. Clyde and P. Muller and G. Parmigiani. Case Studies in Bayesian Statistics 2 ( C. Gatsonis, J. Hodges, R. E. Kass, and N. Singpurwalla, eds.). 1995

    1995

  51. [51]

    C. P. Robert and G. Casella. 2004

    2004

  52. [52]

    Linhart and W

    H. Linhart and W. Zucchini. 1986

    1986

  53. [53]

    Balakrishnan and E

    N. Balakrishnan and E. Cramer. 2014

    2014

  54. [54]

    Audet and V

    C. Audet and V. B \' a chard and S. Digabel. Journal of Global Optimization. 2008

    2008

  55. [55]

    Almohaimeed, Bader , journal=

  56. [56]

    2025 , publisher=

    Dhameliya, Vaibhav N and Maurya, Raj Kamal and Bhattacharya, Ritwik , journal=. 2025 , publisher=

    2025

  57. [57]

    2020 , publisher=

    Bhattacharya, Ritwik , journal=. 2020 , publisher=

    2020

  58. [58]

    2024 , publisher=

    Dhameliya, Vaibhav N and Maurya, Raj Kamal and Bhattacharya, Ritwik , journal=. 2024 , publisher=

    2024

  59. [59]

    2025 , publisher=

    Chakraborty, Siddhartha and Bhattacharya, Ritwik and Pradhan, Biswabrata , journal=. 2025 , publisher=

    2025

  60. [60]

    Statistical models and methods for biomedical and technical systems , author =

  61. [61]

    Morabbi, H and Razmkhah, M , year=

  62. [62]

    1956 , publisher=

    Lindley, Dennis V , journal=. 1956 , publisher=

    1956

  63. [63]

    Audet and J

    C. Audet and J. Brimberg and P. Hansen and N. Mladenovi \' c. Management Science , volume =". 2004

    2004

  64. [64]

    Carrabs and J

    F. Carrabs and J. F. Cordeau and G. Laporte. INFORMS Journal on Computing. 2007

    2007

  65. [65]

    Carrizosa and B

    E. Carrizosa and B. Mar\' t ın Barrag\' a n and F. Plastria and D. R. Morales. INFORMS Journal on Computing. 2007

    2007

  66. [66]

    M. A. Lejeune. European Journal of Operational Research. 2006

    2006

  67. [67]

    Zaher, AM and Ismail, MA and Bahaa, MS , journal=

  68. [68]

    Hong and C

    Y. Hong and C. King and Y. Zhang and W. Q. Meeker. Journal of Quality Technology. 2015

    2015

  69. [69]

    Bhattacharya and B

    R. Bhattacharya and B. Pradhan. Quality and Reliability Engineering International. 2018

    2018

  70. [70]

    Roy and B

    S. Roy and B. Pradhan. Quality and Reliability Engineering International. 2017

    2017

  71. [71]

    1969 , publisher=

    Menke, Warren W , journal=. 1969 , publisher=

    1969

  72. [72]

    1983 , publisher=

    Thomas, Marlin U , journal=. 1983 , publisher=

    1983

  73. [73]

    2017 , publisher=

    Roy, Soumya and Pradhan, Biswabrata , journal=. 2017 , publisher=

    2017

  74. [74]

    Zhang and Z

    C. Zhang and Z. Lin and Z. Lin. Lecture Notes in Computer Science. 2005

    2005

  75. [75]

    Zhang and W

    Y. Zhang and W. Q. Meeker. Metrika. 2005

    2005

  76. [76]

    Hansen and N

    P. Hansen and N. Mladenovi \' c and J. A. M. P \' e rez. 4OR. 2008

    2008

  77. [77]

    Balakrishnan and A

    N. Balakrishnan and A. P. Basu. 1995

    1995

This paper was first reviewed by deepseek-v4-flash on August 2, 2026.