A logistic-transform model on the unit interval: Distributional properties and likelihood-based inference

Abstract

We investigate a logistic-transform model on the unit interval, denoted by NULog, obtained by applying the inverse-logit transformation to a logistic location–scale random variable. Specifically, letting Z ∼ Logistic(µ, s) with (µ, s) ∈ R × (0,∞) and defining X = (1 + e−Z)−1 yields a tractable transformation-based model on (0, 1) rather than a fundamentally new stochastic family. We establish boundary and tail asymptotics, stochastic ordering properties, and explicit representations of the hazard rate together with analytically justified shape transitions. A central structural result shows that likelihood based inference for (µ, s) under the unit-interval representation is exactly equivalent, up to a parameter-free Jacobian term, to inference under the classical logistic location–scale family applied to the logit-transformed observations. This equivalence yields explicit score equations, diagonal Fisher information, and standard asymptotic normality of the maximum likelihood estimators. Finite-sample performance is examined through Monte Carlo experiments. We further develop a quantile-regression extension based on the closed form quantile representation, and illustrate the practical behavior of the model using real datasets with standard goodness-of-fit criteria and graphical diagnostics. The proposed unit-interval construction is presented as an analytically transparent and computationally convenient approach for bounded-response modeling, while avoiding claims of a new inferential mechanism beyond the underlying logistic transformation.

Keywords

• Fisher information
• hazard rate
• likelihood equivalence
• quantile regression
• logistic transformation
• unit-interval model
• stochastic ordering

References

1. 1] H.S. Bakouch, T. Hussain, M. Tosic, V.S. Stojanovi and N. Qarmalah, Unit exponential probability distribution: Characterization and applications in environmental and engineering data modeling, Mathematics 11 (19), 4207, 2023. https://doi.org/10.3390/math11194207