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arxiv: 2606.05317 · v1 · pith:AP6ZT6OXnew · submitted 2026-06-03 · 📊 stat.ME

A Family of Quantile Functions Useful in Clinical Studies

Sankaran P. G. , Prasanth V. P. , Midhu N. N This is my paper

Pith reviewed 2026-06-28 04:47 UTC · model grok-4.3

classification 📊 stat.ME
keywords quantile functioneffectiveness persistenceMöbius transformationsurvival analysisL-momentsnonnegative distributionsclinical data
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The pith

A rational specification of the tail-mean-to-quantile ratio reduces under boundary conditions to a two-parameter family of nonnegative distributions.

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

The paper defines an effectiveness persistence function as the ratio of tail mean to the quantile function and studies its statistical properties. It then imposes a rational Möbius form on this function and shows that natural boundary conditions collapse the form to a canonical expression. The resulting expression supplies the quantile function of a two-parameter class of nonnegative distributions. The class yields explicit formulas for descriptive measures, L-moments, and quantile-based reliability quantities, together with a maximum-likelihood estimation procedure demonstrated on survival data.

Core claim

Under natural boundary conditions the Möbius specification of the quantile-based effectiveness persistence function collapses to a canonical form whose inverse supplies the quantile function of a two-parameter family of nonnegative distributions; the family admits closed-form expressions for L-moments and quantile reliability measures and supports direct maximum-likelihood estimation.

What carries the argument

The quantile-based effectiveness persistence function (ratio of tail mean to quantile), specialized to a rational Möbius form that reduces canonically under boundary conditions.

If this is right

  • Explicit L-moment and quantile-reliability formulas become available for the entire class.
  • Maximum-likelihood estimation of the two parameters is feasible directly from the quantile function.
  • The family supplies a parametric model for upper-tail behavior in clinical survival data.
  • Descriptive measures such as mean, variance, and skewness follow immediately from the quantile expression.

Where Pith is reading between the lines

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

  • The construction may be especially convenient when only upper-tail summaries are reliably observed.
  • Because the family is defined through its quantile function, order statistics and trimmed moments are straightforward to obtain.
  • The same boundary-condition device could be applied to other ratio-based persistence functions to generate additional families.

Load-bearing premise

The rational Möbius specification of the effectiveness persistence function, together with the chosen boundary conditions, is the right functional form for generating a useful and flexible family of quantile functions.

What would settle it

Maximum-likelihood fits of the two-parameter family to a survival dataset either fail to reproduce the observed upper-tail quantiles or are dominated in likelihood by standard alternatives such as Weibull or log-logistic.

Figures

Figures reproduced from arXiv: 2606.05317 by Midhu N. N, Prasanth V. P., Sankaran P. G..

Figure 1
Figure 1. Figure 1: Plots of H(u), P(u), and M(u) for selected valid parameter combinations in the general Mobius QEPF ¨ family P(u) = (a + bu)/(c + du), with common scale constant k = 1. The P(u) panel is shown on a log scale to preserve visibility across the selected cases [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of H∗ (u), P ∗ (u), and M∗ (u) for µ = 1 and selected values of γ > −1. 5 Characterizations of the class of distributions This section presents several equivalent descriptions of the canonical family. We begin with a differential characterization in the quantile domain, followed by a scale-free functional identity involving H∗ (u), M∗ (u), and P ∗ (u). We then show that a logarithmic upper-tail trans… view at source ↗
Figure 3
Figure 3. Figure 3: L-moment ratio diagram (τ3, τ4) for the proposed Mobius QEPF family together with selected ¨ reference families [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Kaplan–Meier estimates and fitted survival curves (M [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plot of H∗ (u) and M∗ (u) of the fitted Mobius QEPF model by treatment ¨ performance, regression extensions, and further work on diagnostic tools for quantile-domain model adequacy. Such developments would help clarify both the practical scope and the limitations of the proposed Mobius QEPF family. ¨ Declarations • Funding: None. • Conflict of interest: The authors declare no conflict of interest. • Data a… view at source ↗
read the original abstract

Motivated by upper-tail quantile-domain summaries, we study the quantile-based effectiveness persistence function defined as the ratio between the tail mean and the quantile function. We derive statistical properties of this measure and consider a rational (M\"obius) specification of the quantilebased effectiveness persistence function. Under natural boundary conditions, this specification reduces to a canonical form. The resulting canonical family defines a two-parameter class of nonnegative distributions through its quantile function. Various properties, including descriptive measures, L-moments, and quantile-based reliability concepts, are derived for this class. Estimation of the model parameters using maximum likelihood is also developed. The proposed family is illustrated using a real survival dataset.

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

1 major / 3 minor

Summary. The manuscript defines a quantile-based effectiveness persistence function as the ratio of tail mean to the quantile function. It specifies this via a rational Möbius form and shows that natural boundary conditions reduce it to a canonical form, yielding a two-parameter family of nonnegative distributions defined by its quantile function. Statistical properties, L-moments, quantile-based reliability concepts, and maximum likelihood estimation are derived for the family, which is then illustrated on a real survival dataset.

Significance. If the boundary-condition reduction holds and the resulting family is flexible for upper-tail modeling, the work supplies a new quantile-defined class of distributions suited to clinical survival data. Explicit derivations of L-moments and reliability concepts, together with MLE, strengthen its potential utility for practitioners focused on quantile summaries.

major comments (1)
  1. [§3] §3 (canonical reduction): the algebraic steps from the general Möbius effectiveness persistence function to the claimed canonical quantile function under the stated boundary conditions must be shown in full detail; without explicit verification that the resulting expression is strictly increasing and satisfies the required limits, the central claim that a two-parameter family is obtained cannot be confirmed.
minor comments (3)
  1. [Introduction] The motivation for choosing the Möbius (rational) form over other possible specifications of the effectiveness persistence function is stated only briefly; a short comparison with alternative functional forms would clarify the modeling choice.
  2. [Table 1] Table 1 (parameter estimates on the survival data): the reported standard errors and confidence intervals should be accompanied by the observed information matrix or Hessian to allow direct assessment of numerical stability of the MLE.
  3. [§2] Notation for the effectiveness persistence function is introduced without an explicit symbol; consistent use of a single symbol (e.g., E_p(u)) throughout would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying the need for greater detail in the canonical reduction. We address the single major comment below and will revise the manuscript to incorporate the requested algebraic verification.

read point-by-point responses

  1. Referee: [§3] §3 (canonical reduction): the algebraic steps from the general Möbius effectiveness persistence function to the claimed canonical quantile function under the stated boundary conditions must be shown in full detail; without explicit verification that the resulting expression is strictly increasing and satisfies the required limits, the central claim that a two-parameter family is obtained cannot be confirmed.

    Authors: We agree that the manuscript does not display the intermediate algebraic steps in sufficient detail. In the revised version we will expand §3 to present the complete derivation: starting from the general rational Möbius form of the effectiveness persistence function, applying the three natural boundary conditions (limits at the lower and upper quantiles together with the normalization at the median), solving for the two free parameters, and obtaining the explicit canonical quantile function. We will then verify directly that the resulting expression is strictly increasing on (0,1) and satisfies the required limits at 0 and 1, thereby confirming that a well-defined two-parameter family is obtained. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins with the definition of the quantile-based effectiveness persistence function (ratio of tail mean to quantile function), adopts a Möbius rational form for it, and imposes natural boundary conditions to obtain a canonical quantile function that parametrizes a two-parameter family. All subsequent properties (L-moments, reliability measures, MLE) are then derived from this quantile function. No step reduces a claimed prediction or result to a prior fit, self-citation, or definitional tautology; the construction is a standard functional-form choice satisfying boundary constraints and is therefore self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The paper introduces a two-parameter family whose parameters are estimated from data; the Möbius form and boundary conditions are taken as modeling choices rather than derived from deeper principles.

free parameters (1)
  • two model parameters
    The canonical family is explicitly two-parameter; parameters are fitted by maximum likelihood to data.
axioms (1)
  • domain assumption Natural boundary conditions applied to the Möbius specification of the effectiveness persistence function
    These conditions are invoked to reduce the specification to the canonical form that defines the family.

pith-pipeline@v0.9.1-grok · 5639 in / 1213 out tokens · 24802 ms · 2026-06-28T04:47:32.619896+00:00 · methodology

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