Pith. sign in

REVIEW 3 major objections 4 minor 32 references

Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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

Pith's one-line read The paper establishes closed-form mgfs and recurrences for HLG generalized order statistics, then gives Bayes estimators under three losses.

desk verdict The generalized-order-statistics moment formulas are a legitimate but routine extension of Liu-Balakrishnan; the Bayesian half is internally inconsistent and its numerical claims are not reproducible as printed. read the letter →

arxiv 2502.01255 v1 pith:7XHG4TFY submitted 2025-02-03 stat.ME math.STstat.TH

Neetu Gupta , S. K. Neogy , Qazi J. Azhad , Bhagwati Devi This is my paper · ORCID
classification stat.MEmath.STstat.TH MSC 62C1062F1062F15
keywords halflogisticgeometricdistributiongeneralizedorderstatisticsmomentgeneratingfunctionrecurrencerelationsBayesianestimationLindleyapproximationMarkovchainMonteCarlo
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 aims to make the half logistic geometric (HLG) distribution—a flexible lifetime model with increasing failure rate—tractable in ordered-data settings. It derives closed-form expressions for the marginal and joint moment generating functions of generalized order statistics (gos), a framework that includes ordinary order statistics, record values, and progressively censored samples as special cases. From those mgfs it obtains single and product moments and recurrence relations, and it shows that when the gos parameters reduce to order statistics the formulas match earlier results for HLG order statistics. The second half of the paper builds approximate Bayes estimators of the shape parameter under squared-error, LINEX, and general-entropy losses using Lindley's approximation and Markov chain Monte Carlo, and applies them to COVID-19 mortality and traction-motor lifetime data.

What carries the argument

The central object is the HLG distribution's survival-function identity f(x) = \bar F(x) - ((2-θ)/2)\bar F(x)^2, with \bar F(x) = 1-F(x). Substituting this identity into the gos density (governed by the parameters m, k and the gamma coefficients γ_r = k + n - r + M_r) splits every moment integral into two $\beta$-family integrals, producing the closed-form mgfs. The joint mgf additionally converts an inner integral into a Gauss hypergeometric function via an incomplete-$\beta$ integral identity and then expands it termwise. Recurrence relations follow from integration by parts on the two terms of the same identity. The incomplete $\beta$ function with upper limit 1-θ/2 is the workhorse of the marginal formulas.

What would settle it

Take n=2, m=0, k=1 and any fixed x_1 < x_2; multiply the likelihood factor (4.1) by the prior (4.5) and simplify at two values of θ. If the resulting ratio differs from the ratio given by the printed posterior (4.6), the Lindley and MCMC numbers in the tables are not posterior estimates under the stated model.

Watch

Extended reading notes

Core claim

Under the generalized order statistics (gos) setup, the paper proves explicit closed-form expressions for the marginal mgf and the joint mgf of the HLG distribution, written as finite sums of incomplete beta functions and an infinite hypergeometric-series sum. It also proves recurrence relations for these mgfs and, by differentiating at zero, for the single and product moments. In the special case m=0, k=1, the gos reduce to ordinary order statistics and the formulas reproduce the known HLG order-statistics moments. For inference, the paper writes the gos likelihood, adopts a gamma prior on the shape parameter, and obtains approximate Bayes estimators under squared-error, LINEX, and general-entropy losses via Lindley's approximation and an MCMC scheme; the order-statistics submodel is used for a simulation study and two real data applications.

Load-bearing premise

The Bayesian estimates stand or fall on the printed posterior density (4.6) being the true product of the gos likelihood (4.1) and the gamma prior, yet the θ-exponents written there do not obviously match the likelihood factors.

Editorial extensions

If this is right

  • Explicit mgfs make single and product moments of gos from the HLG distribution available without numerical integration for any m ≥ -1, k ≥ 1.
  • The recurrence relations allow higher-order moments to be computed from lower-order ones, which reduces computational cost for large n.
  • When m=0 and k=1 the formulas reduce to ordinary order statistics and reproduce the known HLG results, validating the derivation.
  • The three Bayes estimators give practitioners symmetric and asymmetric loss options in lifetime-data analysis.
  • Because gos include record values and progressively Type-II censored order statistics, the derived formulas extend to those sampling schemes as well.

Reading between the lines

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

  • If the printed posterior density is corrected to match the likelihood, the numerical tables for Lindley and MCMC estimates would need to be regenerated; the qualitative comparison of losses may survive, but the printed numbers are not trustworthy as posterior quantities.
  • The same 'express f as a polynomial in the survival function' proof device should yield closed-form mgfs for other distributions whose density admits such a relation, such as other geometric-mixture or exponentiated models.
  • Differentiating the joint mgf at zero beyond first order could produce L-moments or product L-moments, extending the paper's moment toolbox to applications in robust estimation and distribution comparison.
  • The mgf expressions might be inverted numerically to approximate the gos density itself, though the paper does not attempt this.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the half logistic geometric (HLG) distribution under generalized order statistics (gos). It claims closed-form expressions for the marginal mgf (Theorem 2.1) and joint mgf (Theorem 3.1), recurrence relations for single and product moments (Theorems 2.2 and 3.2), and reduction to order statistics that agrees with Liu and Balakrishnan (2020). The second half develops approximate Bayes estimators of the parameter θ under squared error, LINEX, and general entropy losses using Lindley's approximation and MCMC, and reports simulations and two real-data analyses.

Significance. If correct, the moment-generating-function part would provide a useful unification of existing order-statistics results for the HLG distribution, and the reduction to Liu and Balakrishnan (2020) gives independent support for the mgf derivations. However, the Bayesian part, which is a substantial component of the paper, is built on an incorrect posterior density. As printed, the simulation study and the data analyses do not estimate the model that is stated, so the reported superiority of Lindley over MCMC and of GE loss is not supported.

major comments (3)
  1. [Section 4, Eq. (4.6)] The posterior density in (4.6) does not follow from the generalized order statistics likelihood (4.1). From (4.1), the first n−1 factors contribute D_i^{-(m+2)} with D_i = θ + (2−θ)e^{-x_i}, the nth factor contributes D_n^{-(k+1)}, and the Gamma prior in (4.5) contributes θ^{a−1}; multiplying gives a posterior with D_i^{-(m+2)} for i<n, D_n^{-(k+1)}, and θ^a e^{-bθ} (apart from the further issue that (4.1) itself omits the θ^{n−1} factor that should arise from the n densities f(x_i)). Equation (4.6), which prints D_i^{-(m+2)(k+1)} for every i, is therefore a different posterior. Even in the order-statistics case m=0, k=1 used throughout the tables, (4.6) gives exponent −4 per factor while the correct HLG likelihood has exponent −2. Because (4.10), the MCMC algorithm, and all entries in Tables 1–3 and 5–7 are based on (4.6), the Bayesian estimates and subsequent comparisons are not estimates of the stated model.
  2. [Section 4, Eq. (4.10)] The derivatives of the log-likelihood used for Lindley's approximation are not consistent with the stated model. For the likelihood (4.1), the second derivative of log L with respect to θ is −1/θ² + (m+2)Σ_{i=1}^{n−1}(1−e^{-x_i})²/D_i² + (k+1)(1−e^{-x_n})²/D_n², not the expression printed in (4.10). If instead one uses the full gos joint density (1.1), the θ factor appears n times, giving −n/θ² in the second derivative; either way the printed formula has an incorrect last term, whose denominator also uses x_i in place of x_n. The third derivative has the same problems in sign and indexing. Since Lindley's approximation is defined through these derivatives, the Lindley estimates in Section 5 are not reproducible from the stated model.
  3. [Theorems 2.1 and 3.1, and Eqs. (2.4)–(2.5)] The beta function appearing in the mgf formulas is never defined. Theorems 2.1 and 3.1 use B(1−θ/2, a, b), which is evidently intended to be the incomplete beta function B_x(a,b) with x = 1−θ/2, but no definition is given. More seriously, (2.4) and (2.5) write the upper limit as 1−1/θ, which is negative for all θ ∈ (0,1). As printed, the moment expressions obtained by differentiating the mgf are not well defined or computable. The definition of B_x(a,b) and correction of the upper limit to 1−θ/2 are necessary for the moment results.
minor comments (4)
  1. [Section 2, Theorem 2.2] The notation M^p_{r,n,m,k}(t) in (2.6) and then µ^p in (2.10) is not defined; it is unclear whether p denotes a moment order or an order of differentiation, and the relation between (2.6) and the subsequent moment recurrence should be stated explicitly.
  2. [Section 3, Theorem 3.2] The proof refers to 'making use of (1.9)', but the identity f(x) = bar{F}(x) − ((2−θ)/2) bar{F}(x)^2 is equation (1.8); the reference should be corrected.
  3. [Section 5, Table 3] Some entries, such as the MSE of 0.000012 for θ=0.6, n=10, Prior II, GE loss with c=0.5, appear implausibly small relative to neighboring entries and should be checked against the corrected posterior computations.
  4. [Throughout] There are numerous typographical errors in subscripts and exponents, including the use of x_i in the denominator of the last term in (4.10) and the unreadable exponent in (4.6). A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the generalized order statistics moment derivations are self-contained from the HLG survival identity and reduce to Liu–Balakrishnan only as a consistency check; the posterior inconsistency in (4.6) is a correctness defect, not a circularity.

full rationale

No load-bearing circular step can be exhibited. Theorems 2.1 and 3.1 start from the standard generalized order statistics densities (1.2)-(1.3) and the identity (1.8), which itself follows from the stated HLG cdf and pdf in (1.6)-(1.7). The mgf integrals reduce to incomplete beta functions with upper limit 1 - theta/2 by a direct substitution, so the formulas are fresh derivations rather than restatements of the target moments. The recurrence relations (2.6) and (3.6) follow by integration by parts and differentiation of the same identities, and the m = 0, k = 1 reductions agree with Liu and Balakrishnan (2020) as a consistency check, not as an input to the proof. The self-citations to Gupta and Jamal (2019), Azhad et al. (2020), Arshad et al. (2021, 2022), and Albalawi et al. (2022) are contextual or algorithmic references and do not supply the load-bearing steps. The Bayesian estimators are obtained from a stated likelihood and prior; theta is not fitted to the quantity being reported, so no fitted input is renamed as a prediction. Separately and outside the circularity question, Eq. (4.6) does not follow from likelihood (4.1) and prior (4.5): multiplying the first n-1 factors, whose denominator exponent is -(m+2), the nth factor, whose denominator exponent is -(k+1), and the Gamma prior theta^{a-1} e^{-b theta} gives no common factor exponent -(m+2)(k+1) and a different power of theta. This is an internal correctness problem for the numerical Bayesian results in Sections 5-6, but it is not a circularity and is not part of the derivation chain that produces the moment formulas.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The core moment derivations introduce no new entities. The numerical inference depends on chosen priors, loss constants, and unreported MCMC settings. The main unstated mathematical assumption is the meaning of B as an incomplete beta function.

free parameters (3)
  • Gamma prior hyperparameters a,b = a=2, b=1 and b=2 in simulation and applications
    Chosen by hand; MSE rankings of estimators depend on these values, so they are not part of the theoretical derivation.
  • Loss function shape parameters c = c=-0.5, 0.5, 1, 1.5
    Selected for LINEX and GE loss functions; conclusions about which loss performs best are conditional on this set.
  • MCMC tuning parameters = not reported
    Metropolis-Hastings proposal variance, chain length T, and burn-in are not specified, yet the simulation results depend on them.
assumptions (4)
  • domain assumption Generalized order statistics joint density (1.1) from Kamps (1995) is the correct sampling model.
    All moment formulas and the likelihood start from this density.
  • standard math The relation f(x) = survival(x) minus ((2-theta)/2) survival(x)^2 holds for the HLG distribution.
    Stated in (1.8) and used in the recurrence relations; it follows algebraically from (1.6) and (1.7).
  • standard math Incomplete beta and Gauss hypergeometric identities, including the Dutka (1981) integral (3.4), are valid and can be interchanged with infinite sums.
    Used in Theorems 2.1 and 3.1; no convergence justification is given for expanding the hypergeometric function and interchanging summation and integration.
  • domain assumption Lindley approximation and MH-MCMC produce reliable posterior summaries for this low-dimensional problem.
    No convergence diagnostics, standard errors, or tuning details support this for the present posterior.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics." pith.science (2026). https://pith.science/paper/7XHG4TFY

@misc{pith2026250201255,
  author       = {Pith},
  title        = {Pith review of: Inference of Half Logistic Geometric Distribution Based on Generalized Order Statistics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7XHG4TFY}},
  note         = {Machine review of arXiv:2502.01255}
}
abstract

As the unification of various models of ordered quantities, generalized order statistics act as a simplistic approach introduced in \cite{kamps1995concept}. In this present study, results pertaining to the expressions of marginal and joint moment generating functions from half logistic geometric distribution are presented based on generalized order statistics framework. We also consider the estimation problem of $\theta$ and provides a Bayesian framework. The two widely and popular methods called Markov chain Monte Carlo and Lindley approximations are used for obtaining the Bayes estimators.The results are derived under symmetric and asymmetric loss functions. Analysis of the special cases of generalized order statistics, \textit{i.e.,} order statistics is also presented. To have an insight into the practical applicability of the proposed results, two real data sets, one from the field of Demography and, other from reliability have been taken for analysis.

Figures

Figures reproduced from arXiv: 2502.01255 by the authors.

Figure 1
Figure 1. Trace plots of θ 6 Data Analysis Application 6.1. In order to exhibit the application of proposed results in the field of demo￾graphic study, we have considered mortality rates of 30 days of Netherlands country due to COVID-19 from March 31, 2020 to April 30, 2020. The COVID-19 is the third-highest cause of deaths in 2020 which has been revealed by the US Centers for Disease Control and Prevention (CDC). The mortali… view at source ↗
Figure 2
Figure 2. Fitting of Mortality Data with HLG Distribution [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Fitting of age for traction motors data with HLG Distribution [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 31 canonical work pages

  1. [1]

    Aboeleneen, Z. (2010). Inference for weibull distribution under generalized order statistics. Mathematics and Computers in Simulation , 81(1):26--36

    work page 2010

  2. [2]

    Akhter, Z., Saran, J., Verma, K., and Pushkarna, N. (2020). Moments of order statistics from length-biased exponential distribution and associated inference. Annals of Data Science , pages 1--26

    work page 2020

  3. [3]

    C., Azhad, Q

    Albalawi, O., Kabdwal, N. C., Azhad, Q. J., Hora, R., and Alsaedi, B. S. (2022). Estimation of the generalized logarithmic transformation exponential distribution under progressively type-ii censored data with application to the covid-19 mortality rates. Mathematics , 10(7):1015

    work page 2022

  4. [4]

    F., Athar, H., and Fawzy, M

    Alharbi, Y. F., Athar, H., and Fawzy, M. A. (2021). Moment properties of generalized order statistics from exponentiated generalized class of distributions. Thailand Statistician , 19(4):797--811

    work page 2021

  5. [5]

    M., Badr, M

    Almarashi, A. M., Badr, M. M., Elgarhy, M., Jamal, F., and Chesneau, C. (2020). Statistical inference of the half-logistic inverse rayleigh distribution. Entropy , 22(4):449

    work page 2020

  6. [6]

    M., Rezk, H

    Alotaibi, R. M., Rezk, H. R., Ghosh, I., and Dey, S. (2020). Bivariate exponentiated half logistic distribution: Properties and application. Communications in Statistics-Theory and Methods , pages 1--23

    work page 2020

  7. [7]

    Azhad, Q., Gupta, N., and Pathak, A

    Arshad, M., J. Azhad, Q., Gupta, N., and Pathak, A. K. (2021). Bayesian inference of unit gompertz distribution based on dual generalized order statistics. Communications in Statistics-Simulation and Computation , pages 1--19

    work page 2021

  8. [8]

    Arshad, M., Khandelwal, N., and Azhad, Q. J. (2022). Stress--strength reliability estimation in a system with p-type non-identical multicomponents from prhr family based on records. Journal of Statistical Computation and Simulation , pages 1--32

    work page 2022

Show all 32 references
  1. [9]

    J., Arshad, M., and Khandelwal, N

    Azhad, Q. J., Arshad, M., and Khandelwal, N. (2022). Statistical inference of reliability in multicomponent stress strength model for pareto distribution based on upper record values. International Journal of Modelling and Simulation , 42(2):319--334

    2022

  2. [10]

    J., Arshad, M., and Misra, A

    Azhad, Q. J., Arshad, M., and Misra, A. K. (2020). Estimation of common location parameter of several heterogeneous exponential populations based on generalized order statistics. Journal of Applied Statistics , pages 1--18

    2020

  3. [11]

    Balakrishnan, N. (1985). Order statistics from the half logistic distribution. Journal of Statistical Computation and Simulation , 20(4):287--309

    1985

  4. [12]

    and Saleh, H

    Balakrishnan, N. and Saleh, H. (2013). Recurrence relations for single and product moments of progressively type-ii censored order statistics from a generalized half-logistic distribution with application to inference. Journal of Statistical Computation and Simulation , 83(9):...

    2013

  5. [13]

    Burkschat, M. (2009). Linear estimators and predictors based on generalized order statistics from generalized pareto distributions. Communications in Statistics-Theory and Methods , 39(2):311--326

    2009

  6. [14]

    Dutka, J. (1981). The incomplete beta function—a historical profile. Archive for history of exact sciences , pages 11--29

    1981

  7. [15]

    D., and Wilson, S

    Eliashberg, J., Singpurwalla, N. D., and Wilson, S. P. (1997). Calculating the reserve for a time and usage indexed warranty. Management Science , 43(7):966--975

    1997

  8. [16]

    B., Stern, H

    Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian data analysis . CRC press

    2013

  9. [17]

    Gumbel, E. J. (1944). Ranges and midranges. The Annals of Mathematical Statistics , 15(4):414--422

    1944

  10. [18]

    and Jamal, Q

    Gupta, N. and Jamal, Q. A. (2019). Inference for weibull generalized exponential distribution based on generalized order statistics. Journal of Applied Mathematics and Computing , 61(1):573--592

    2019

  11. [19]

    Jeon, Y. E. and Kang, S.-B. (2020). Estimation for the half-logistic distribution based on multiply type-ii hybrid censoring. Physica A: Statistical Mechanics and its Applications , 550:124501

    2020

  12. [20]

    and Jim \'e nez-Gamero, M

    Jodr \'a , P. and Jim \'e nez-Gamero, M. (2014). On a logarithmic integral and the moments of order statistics from the weibull-geometric and half-logistic families of distributions. Journal of Mathematical Analysis and Applications , 410(2):882--890

    2014

  13. [21]

    and Bai, D

    Jung, M. and Bai, D. S. (2007). Analysis of field data under two-dimensional warranty. Reliability Engineering & System Safety , 92(2):135--143

    2007

  14. [22]

    Kamps, U. (1995). A concept of generalized order statistics. Journal of Statistical Planning and Inference , 48(1):1--23

    1995

  15. [23]

    Khatoon, B., JS Khan, M., Sharma, A., and AR Khan, M. (2021). On moments and entropy of kumaraswamy power function distribution based on generalized order statistics with its application. Journal of Statistics Applications & Probability , 10(3):743--758

    2021

  16. [24]

    and Han, K

    Kim, C. and Han, K. (2014). Bayesian estimation of rayleigh distribution based on generalized order statistics. Applied Mathematics Sciences , 8(150):7475--7485

    2014

  17. [25]

    Lindley, D. (1980). Approximate bayesian method, trabajos de estadistica, 31, 223--237

    1980

  18. [26]

    and Balakrishnan, N

    Liu, K. and Balakrishnan, N. (2020). Recurrence relations for moments of order statistics from half logistic-geometric distribution and their applications. Communications in Statistics-Simulation and Computation , pages 1--19

    2020

  19. [27]

    Safi, S. K. and Al Sheikh Ahmed, R. H. (2013). Generalized order statistics from generalized exponential distributions in explicit forms. Electronic Journal of Applied Statistical Analysis , 6(2)

    2013

  20. [28]

    Singh, B., Khan, R., and Khan, M. A. (2021). Exact moments of generalized pareto distribution based on generalized order statistics and characterizations. Pakistan Journal of Statistics and Operation Research , pages 213--226

    2021

  21. [29]

    Verhulst, P. (1845). La loi d’accroissement de la population. Nouv. Mem. Acad. Roy. Soc. Belle-lettr. Bruxelles , 18(1):1--38

  22. [30]

    Verhulst, P.-F. \ c c . o. (1838). Notice on the law that the population follows in its growth. Corresp. Math. Phys. , 10:113--126

  23. [31]

    and Jose, J

    Xavier, T. and Jose, J. K. (2020). A study of stress-strength reliability using a generalization of power transformed half-logistic distribution. Communications in Statistics-Theory and Methods , pages 1--17

    2020

  24. [32]

    and Gui, W

    Zhang, F. and Gui, W. (2020). Parameter and reliability inferences of inverted exponentiated half-logistic distribution under the progressive first-failure censoring. Mathematics , 8(5):708

    2020

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.