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 $Z \sim \mathrm{Logistic}(\mu, s)$ with $(\mu, s) \in \mathbb{R} \times (0, \infty)$ 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 $(\mu, s)$ 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

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