- Article
33 Pages
In this study, we introduce and study the record-based transmuted Pareto distribution (RBTPD), a three-parameter model obtained by applying the record-based transmutation map to the classical two-parameter Pareto distribution. Its relationship to the record-based transmuted power Lomax distribution is set out explicitly, and the contribution of this paper is located in the Pareto parameterization itself: the closed-form properties it permits, the fact that its lower endpoint is a threshold parameter with a substantive interpretation and requires non-regular inference, and a systematic comparison of five estimation methods. The additional parameter is the weight attached to the second upper record value of the baseline; it is confined to the unit interval and returns the Pareto distribution at zero. The proposed model is more flexible than the baseline and, unlike several existing extensions, has a density that remains bounded on the whole support. We derive the distribution; density; survival and hazard rate functions; the quantile function, in terms of the Lambert W function; moments and incomplete moments; mean deviations; Bonferroni and Lorenz curves; order statistics; record values; and Renyi and Shannon entropies. Five estimation methods are compared in a Monte Carlo study: maximum likelihood, least squares, weighted least squares, Anderson–Darling, and maximum product of spacings. For α and for p away from the boundaries of its range, the mean squared errors of all five estimators decrease with increasing sample size, a pattern consistent with consistency, though a finite range of sample sizes cannot establish it; the threshold is a support-boundary parameter and is treated separately, as the model is non-regular in it. Maximum likelihood performs best overall and is markedly the most stable numerically. The model is fitted to two real data sets, one medical and one hydrological, and gives a better fit than both the baseline Pareto and the transmuted Pareto distributions on every criterion considered. Compared with competing models outside the Pareto family, the proposed model gives generally comparable results and performs relatively well in some cases. Bootstrap-calibrated goodness-of-fit tests indicate adequate absolute fit for the medical data but not for the hydrological series, and the conditions under which the model is appropriate are stated explicitly.
Axioms
22 October 2026
![(a) The metric coefficients
f
(
r
)
(in blue) and
h
(
r
)
(in red) for the non-Schwarzschild solution with
r
0
=
1
and
α
=
1
/
2
obtained in Ref. [95], together with the corresponding analytic approximations obtained in [96] (thick dotted curves). Also shown are the Schwarzschild metric coefficients (dot-dashed curves). For clarity
h
(
r
)
and its corresponding Schwarzschild coefficient has been rescaled to approach 3/4 instead of 1 to avoid overlap in the asymptotic region. (b) The mass of the non-Schwarzschild black holes (solid curves) and the corresponding Schwarzschild black hole (dashed line) as a function of
r
0
, for
α
=
1
/
3
(green),
α
=
1
/
2
(red),
α
=
3
/
4
(blue).](https://mdpi-res.com/axioms/axioms-15-00610/article_deploy/html/images/axioms-15-00610-g001-550.jpg)
