Modified New [0, 1] – Right Truncated Exponential Rayleigh Distribution Structure and Application

Overview

The paper introduces a right-truncated [0,1] exponential-Rayleigh lifetime distribution ([0,1]-RTERD) with two parameters (scale and shape) intended for data constrained to the unit interval. The manuscript develops the model’s basic functional forms—probability density, cumulative, survival, and hazard functions—and reports additional derived properties including median, origin moments, skewness and kurtosis measures, order-statistic behavior, moment-generating considerations, Rényi entropy, and the quantile function. Graphical representations of these functions are provided to illustrate shape control induced by the truncation and normalization.

Methods and approach

Model construction follows truncation and normalization of an exponential-Rayleigh baseline to the [0,1] interval, yielding a two-parameter family. Statistical characterizations are obtained by explicit mathematical construction and plotted for parameter regimes of interest. Parameter estimation is performed via maximum likelihood estimation (MLE). Model comparison against alternative continuous distributions uses statistical information criteria to evaluate relative fit. Applications to empirical lifespan and reliability datasets are used to assess empirical performance.

Key Findings

The paper presents expressions for the distribution’s PDF, CDF, survival and hazard functions and computes several descriptive and inferential quantities analytically where derivation is possible; the manuscript does not universally claim closed-form expressions for all quantities. Maximum likelihood estimates of the two parameters are reported for applied datasets, and the authors compare information-criterion values between the [0,1]-RTERD and competing continuous models. In the reported examples, the [0,1]-RTERD frequently attains better (lower) information-criterion scores relative to some traditional models, suggesting improved fit in those cases.

Implications

The [0,1]-RTERD offers a structured alternative for modeling lifetime or reliability outcomes confined to [0,1], with flexible shape, center, and scale behavior introduced by truncation and normalization. The analytic constructions and graphical illustrations clarify how parameter changes affect moments and instantaneous measures such as the hazard. Empirical comparisons in the manuscript indicate that the model can provide competitive or superior information-criterion performance for certain datasets, warranting further evaluation in domain-specific applications rather than blanket prescriptive adoption.