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arxiv: 2603.27323 · v2 · submitted 2026-03-28 · 🧮 math.ST · stat.TH

Property Of The Beta Modified Weibull Distribution With Six Parameters

Didier Alain Njamen Njomen , Fidel Djongreba Ndikwa This is my paper

Pith reviewed 2026-05-14 21:57 UTC · model grok-4.3

classification 🧮 math.ST stat.TH MSC 62E1060E05
keywords Beta Modified Weibullsix-parameter distributionhazard function shapesgeneralized distributiondensity functionsurvival analysisWeibull distributionbeta distribution
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The pith

A six-parameter Beta Modified Weibull distribution reproduces every density and hazard shape from prior literature.

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

The paper defines a new six-parameter Beta Modified Weibull distribution and derives its cumulative distribution, survival, probability density, and hazard functions. It shows that this distribution includes many existing distributions as special cases through restrictions on the parameters. Numerical simulations illustrate that the density function can match all shapes appearing in earlier work, while the hazard function can be constant, increasing, decreasing, U-shaped, or inverted U-shaped. These results indicate that the new family encompasses the flexibility of previous Weibull-based models.

Core claim

The six-parameter Beta Modified Weibull distribution is constructed such that its density and hazard functions encompass all shapes present in the literature on Weibull-based distributions, achieved by varying the six parameters appropriately, and it includes numerous sub-distributions that match those previously proposed.

What carries the argument

The six-parameter Beta Modified Weibull distribution, which extends the Weibull by incorporating beta-distribution scaling and additional modification parameters to control the shapes of the density and hazard functions.

Load-bearing premise

That appropriate choices of the six parameter values can achieve every possible shape without gaps or impossible regions in the parameter space.

What would settle it

Finding a specific hazard shape, such as a multimodal hazard or a particular bathtub curve variant, that cannot be obtained for any positive real values of the six parameters.

Figures

Figures reproduced from arXiv: 2603.27323 by Didier Alain Njamen Njomen, Fidel Djongreba Ndikwa.

Figure 1
Figure 1. Figure 1: Plots of the modified Beta weibull density for some parameter values. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the Modified Beta Weibull hazard function for some parameter values [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

The aim of this article is to determine a new six-parameter Beta Weibull distribution and its various associated functions, namely the cumulative distribution, survival, probability density and hazard functions. Next, we determine the sub-distributions of the new distribution and show that the latter generalizes those of the literature. Finally, numerical simulations were performed and show that the shapes of the density function of the new distribution cover all those in the literature, and the shapes of hazard functions (constant, increasing, decreasing, $\bigcup$-shaped and $\bigcap$-shaped) are represented in the new distribution and encompass all existing distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a new six-parameter Beta Modified Weibull distribution, derives its CDF, PDF, survival function, and hazard function via standard transformations, identifies sub-distributions that recover several existing Weibull-based families, and uses numerical simulations with selected parameter values to illustrate that the resulting density and hazard shapes (constant, monotone, ∪-shaped, ∩-shaped) encompass all shapes appearing in the prior literature.

Significance. A rigorously established six-parameter family that provably recovers all prior shapes without gaps in the parameter space would supply a single flexible model for lifetime data exhibiting arbitrary hazard behaviors. The current manuscript supplies routine derivations and illustrative plots but does not yet deliver the required analytic characterization of the shape map.

major comments (2)
  1. [Numerical simulations section] The central claim that the new distribution's density and hazard shapes 'cover all those in the literature' and 'encompass all existing distributions' rests exclusively on exhibiting a handful of chosen six-tuples that produce the listed shapes. No derivation is given of the conditions on the six parameters that govern the number of turning points or monotonicity of the hazard rate, so it remains unproven that the image of the shape map is surjective onto every previously observed shape.
  2. [Sub-distributions section] The statement that the six-parameter model generalizes the distributions of the literature is asserted by direct substitution in the simulations, but the explicit algebraic reductions (e.g., the precise values or limits of the extra parameters that recover each named sub-distribution) are not derived or tabulated.
minor comments (2)
  1. [Abstract] The abstract and introduction refer to 'all those in the literature' without citing the specific prior distributions whose shapes are being recovered.
  2. [Introduction] Parameter names and domains are introduced only after the CDF expression, making the early discussion of reductions difficult to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will make targeted revisions to strengthen the presentation of sub-distributions and to moderate the claims about shape coverage while preserving the illustrative value of the simulations.

read point-by-point responses

  1. Referee: [Numerical simulations section] The central claim that the new distribution's density and hazard shapes 'cover all those in the literature' and 'encompass all existing distributions' rests exclusively on exhibiting a handful of chosen six-tuples that produce the listed shapes. No derivation is given of the conditions on the six parameters that govern the number of turning points or monotonicity of the hazard rate, so it remains unproven that the image of the shape map is surjective onto every previously observed shape.

    Authors: We agree that the manuscript relies on selected numerical examples rather than an exhaustive analytic characterization of turning points and monotonicity regions in the six-dimensional parameter space. The simulations were intended to illustrate that the family can produce the standard shapes (constant, monotone, U-shaped, inverted-U-shaped) reported in the literature. In revision we will (i) rephrase the abstract and simulation section to state that the distribution is capable of generating these shapes, as demonstrated by the chosen parameter sets, and (ii) add a short remark that a complete analytic mapping of the shape space is left for future investigation. Additional simulation runs with varied parameter combinations will be included to broaden the illustration. revision: partial

  2. Referee: [Sub-distributions section] The statement that the six-parameter model generalizes the distributions of the literature is asserted by direct substitution in the simulations, but the explicit algebraic reductions (e.g., the precise values or limits of the extra parameters that recover each named sub-distribution) are not derived or tabulated.

    Authors: We will expand the sub-distributions section with explicit algebraic reductions. A new table will list, for each referenced sub-distribution, the exact values or limiting cases (e.g., setting one or more of the six parameters to 1, 0, or infinity) that recover the target model. These reductions will be derived directly from the CDF or PDF expressions and cross-checked against the original references. revision: yes

Circularity Check

0 steps flagged

No circularity; distribution definition and shape illustrations are independent of inputs

full rationale

The manuscript defines the six-parameter Beta Modified Weibull via a direct functional form, derives the CDF, PDF, survival and hazard functions algebraically from that definition, and then selects specific parameter tuples to generate example plots. No step fits parameters to data and renames the fit a prediction, invokes a self-citation as the sole justification for uniqueness or the ansatz, or reduces the claimed shape coverage to a tautological re-expression of the chosen inputs. The generalization statement follows from the algebraic nesting of sub-distributions, which is verifiable by substitution rather than by construction from the simulation results themselves.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the proposed PDF integrates to one and that numerical parameter sweeps adequately sample the full range of possible shapes.

free parameters (1)
  • six distribution parameters
    The model is defined with six free parameters whose specific values are chosen in simulations to produce desired shapes; no external data fitting is described.
axioms (1)
  • standard math The proposed density function integrates to one over its support for admissible parameter values.
    This is invoked by construction when defining the PDF and deriving the CDF.

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Carrasco JMF, Ortega EMM and Cordeiro GM. (2008). A generalized modified Weibull distribution for lifetime modeling. Computational Statistics and Data Analysis

    work page 2008

  2. [2]

    (2025) Estimation problems based on dependent data

    Djongreba NF. (2025) Estimation problems based on dependent data. Theses, Université de Maroua. https://hal.science/tel-05427772

    work page 2025

  3. [3]

    Eugene N, Lee C and Famoye F. (2002). Beta-normal Distribution and its applications. Com- munication in Statistics - Theory and Methods, 31, 497-512

    work page 2002

  4. [4]

    (2005) The beta Weibull distribution

    Famoye F, Lee C, Olumolade O. (2005) The beta Weibull distribution. Journal of Sta- tistical Theory and Applications; 4(2):121–36

    work page 2005

  5. [5]

    Jeong JH (2006), A new parametric family for modelling cumulative incidence function: application to breast cancer data. J.R. Statistic. Soc. A, 169:289-303

    work page 2006

  6. [6]

    Gupta RD and Kundu D. (1999). Generalized exponential distributions. Austral. New Zeal. J. Statist., 41, 173-188

    work page 1999

  7. [7]

    Gupta RD and Kundu D. (2001). Exponentiated exponential distribution: an alternative to gamma and Weibull distributions. Biometrical Jour., 43, 117-130

    work page 2001

  8. [8]

    The modified beta Weibull distribution, Hacettepe Journal of Math- ematics and Statistics

    Khan MN (2015). The modified beta Weibull distribution, Hacettepe Journal of Math- ematics and Statistics. 44(6):1553-1568. 12

    work page 2015

  9. [9]

    Kundu D and Rakab MZ. (2005). Generalized Rayleigh distribution: different methods of estimation. Computational Statistics and Data Analysis, 49, 187-200

    work page 2005

  10. [10]

    Lai CD, Xie M and Murthy DNP. (2003). A modified Weibull distribution. Transactions on Reliability, 52, 33-37

    work page 2003

  11. [11]

    Lee C, Famoye F and Olumolade O (2007) Beta-Weibull Distribution: Some Properties and Applications to Censored Data, Journal of Modern Applied Statistical Methods: Vol. 6 : Iss. 1 , Article 17

    work page 2007

  12. [12]

    Mudholkar GS and Srivastava DK. (1993). Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Transaction on Reliability, 42, 299-302

    work page 1993

  13. [13]

    Mudholkar GS, Srivastava DK and Friemer M. (1995). The exponentiated Weibull fam- ily: A reanalysis of the bus-motor-failure data. Technometrics, 37, 436-445

    work page 1995

  14. [14]

    A generalization of the Weibull distribution with application to the analysis of survival data

    Mudholkar GS, Srivastava DK and Kollia GD (1996). A generalization of the Weibull distribution with application to the analysis of survival data. J. Amer. Statist. Assoc., 91, 1575-1583

    work page 1996

  15. [15]

    (2004), The beta Gumbel distribution, Mathematical Prob- lems in Engineering, 1, 323-332

    Nadarajah S and Kotz S. (2004), The beta Gumbel distribution, Mathematical Prob- lems in Engineering, 1, 323-332

    work page 2004

  16. [16]

    Nadarajah S and Kotz S. (2006). The beta exponential distribution. Reliability Engi- neering and System Safety, 91, 689-697

    work page 2006

  17. [17]

    Sankhya Ser

    Nadarajah S, Teimouri M and Shih SH.(2014) Modified Beta Distributions. Sankhya Ser. B 76, 19-48

    work page 2014

  18. [18]

    A., Shen, J

    Ngatchou-Wandji, J., Ltaifa, M., Njamen Njomen, D. A., Shen, J. (2022). Nonpara- metric Estimation of the Density Function of the Distribution of the Noise in CHARN Models. Mathematics, 10(4), 624

    work page 2022

  19. [19]

    Density and Risk Function of the Circular Kernel Study, European Journal of Pure and Applied Mathematics, 12(4), 1612-1642

    Njomen DAN, Yayebga HC (2019). Density and Risk Function of the Circular Kernel Study, European Journal of Pure and Applied Mathematics, 12(4), 1612-1642

    work page 2019

  20. [20]

    The beta modified Weibull distribution, Lifetime Data Anal

    Silva GO, Edwin MM, Ortega and Cordeiro GM (2010). The beta modified Weibull distribution, Lifetime Data Anal. 16, 409-430

    work page 2010

  21. [21]

    (2009) A new generalization of Weibull dis- tribution with application to a breast cancer data set

    Wahed AS, Luong TM, and Jeong J-H. (2009) A new generalization of Weibull dis- tribution with application to a breast cancer data set. Statistics in Medicine; 28(16): 2077-2094

    work page 2009

  22. [22]

    Weibull WA. (1951). Statistical distribution function of wide applicability. ASME Journal of Applied Mechanics Transactions of the American Society of Mechanical Engineers:293–7

    work page 1951

  23. [23]

    Zelen M and Dannemiller MC. (1961). The robustness of life testing procedures derived from the exponential distribution. Technometrics 3, 29-49. 13

    work page 1961