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arxiv: 1907.00668 · v1 · pith:IAPOE4T4new · submitted 2019-07-01 · 🧮 math.ST · math.PR· stat.TH

Power Lindley distribution and software metrics

Mohammed Khalleefah , Sofiya Ostrovska , Mehmet Turan This is my paper

Pith reviewed 2026-05-25 11:39 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH
keywords power Lindley distributionmoment determinacymoment indeterminacysoftware metricsstatistical distributionsdata fitting
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The pith

The power Lindley distribution is moment-indeterminate for certain parameter values and determinate for others.

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

The paper establishes conditions on the parameters of the power Lindley distribution under which its moments do or do not uniquely determine the distribution. A sympathetic reader would care because non-uniqueness of the moment problem means multiple distributions can produce the same moment sequence, affecting statistical identification. The results are followed by fits of the distribution to software metrics data sets to show possible uses in practice.

Core claim

The power Lindley distribution is moment-indeterminate when the power parameter lies in specific intervals, shown by constructing distinct distributions that share the same moments, and moment-determinate outside those intervals; the distribution is then applied to model software metrics data.

What carries the argument

The power Lindley distribution, formed by a power transformation of the Lindley distribution, whose moment sequence is analyzed for uniqueness depending on the parameters.

If this is right

  • When the distribution is indeterminate, its moments alone cannot identify it uniquely among all possible distributions.
  • The distribution can still be used to fit positive-valued data such as software metrics.
  • Parameter selection can ensure moment determinacy if uniqueness from moments is needed for inference.

Where Pith is reading between the lines

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

  • The same parameter-based criteria for (in)determinacy might extend to other members of the Lindley family.
  • In applications where moments are matched to data, indeterminacy would require extra conditions such as support restrictions to restore uniqueness.

Load-bearing premise

The power Lindley distribution is assumed to provide a useful description of software metrics data that justifies moving from the moment analysis to the fitting examples.

What would settle it

An explicit demonstration that two different distributions with the claimed identical moments do not exist for the parameter values asserted to be indeterminate would falsify the indeterminacy results.

Figures

Figures reproduced from arXiv: 1907.00668 by Mehmet Turan, Mohammed Khalleefah, Sofiya Ostrovska.

Figure 1
Figure 1. Figure 1: Fitted distributions for DIT-system category [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Fitted distributions for NOC-system category [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Data and fitted densities on different intervals [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
read the original abstract

The Lindley distribution and its numerous generalizations are widely used in statistical and engineering practice. Recently, a power transformation of Lindley distribution, called the power Lindley distribution, has been introduced by M. E. Ghitany et al., who initiated the investigation of its properties and possible applications. In this article, new results on the power Lindley distribution are presented. The focus of this work is on the moment-(in)determinacy of the distribution for various values of the parameters. Afterwards, certain applications are provided to describe data sets of software metrics.

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 / 1 minor

Summary. The manuscript derives new results on the moment-(in)determinacy of the power Lindley distribution across parameter values and applies the distribution to model several software metrics data sets.

Significance. If the moment-indeterminacy results are fully derived with explicit parameter ranges and the data applications include comparative goodness-of-fit statistics, the work would strengthen the literature on generalized Lindley families and their use for positive skewed data in reliability and software engineering.

major comments (2)
  1. [Application section] Application section: the text proceeds directly to fitting examples without reporting comparative goodness-of-fit statistics (AIC, BIC, KS statistic, or likelihood-ratio tests) against the ordinary Lindley, Weibull, gamma, or lognormal distributions on the same data sets; this leaves the claim that the power Lindley yields a meaningfully better or novel description resting on an untested assumption.
  2. [Moment-indeterminacy analysis] Moment-indeterminacy section: the abstract states that new results on moment-(in)determinacy are derived, yet the manuscript does not supply the full derivations or explicit statements of the parameter ranges for which indeterminacy holds, preventing verification of the central analytic claims.
minor comments (1)
  1. Add error bars or standard errors to any fitted parameter estimates and goodness-of-fit values reported in the data examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will revise the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Application section] Application section: the text proceeds directly to fitting examples without reporting comparative goodness-of-fit statistics (AIC, BIC, KS statistic, or likelihood-ratio tests) against the ordinary Lindley, Weibull, gamma, or lognormal distributions on the same data sets; this leaves the claim that the power Lindley yields a meaningfully better or novel description resting on an untested assumption.

    Authors: We agree that direct comparison via standard goodness-of-fit criteria is necessary to substantiate the practical advantage of the power Lindley distribution. In the revised manuscript we will add tables reporting AIC, BIC, Kolmogorov-Smirnov statistics and likelihood-ratio tests for the power Lindley versus the ordinary Lindley, Weibull, gamma and lognormal distributions on each software-metrics data set. revision: yes

  2. Referee: [Moment-indeterminacy analysis] Moment-indeterminacy section: the abstract states that new results on moment-(in)determinacy are derived, yet the manuscript does not supply the full derivations or explicit statements of the parameter ranges for which indeterminacy holds, preventing verification of the central analytic claims.

    Authors: The moment-indeterminacy results form the theoretical core of the paper. We will expand the relevant section (and add an appendix if needed) to present the complete derivations together with explicit statements of the parameter regions (in terms of the two shape parameters) for which the distribution is moment-indeterminate or determinate. revision: yes

Circularity Check

0 steps flagged

No significant circularity; moment results use standard techniques independent of data fits

full rationale

The paper's primary chain derives moment-(in)determinacy results for the power Lindley distribution via analytic methods on the Stieltjes moment problem, citing the distribution's prior introduction by Ghitany et al. (distinct authors) only for background. No equations reduce a claimed prediction to a fitted parameter from the software metrics data, nor do self-citations bear the load of uniqueness or ansatzes. The applications section simply fits the distribution to data sets as descriptive examples, without presenting fitted values as independent predictions. This is self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard results from probability theory concerning the moment problem and on the definition of the power Lindley density; no new entities are postulated and no parameters are fitted inside the moment analysis itself.

axioms (1)
  • standard math The classical moment problem on the positive reals determines uniqueness or non-uniqueness of a distribution from its moment sequence.
    Invoked when the paper classifies the power Lindley distribution as moment-indeterminate for certain parameter values.

pith-pipeline@v0.9.0 · 5617 in / 1192 out tokens · 26351 ms · 2026-05-25T11:39:59.433346+00:00 · methodology

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