A Symmetric Maximum-Entropy Characterisation of the Weibull Distribution via Fractional Moments

The Weibull distribution is one of the most versatile and widely applied continuous probability distributions in reliability engineering, survival analysis, wind-energy modelling, and extreme-value theory. Classical parameter estimation relies on maximum likelihood estimation or method-of-moments using integer-order moments, both of which may suffer from instability or high variance in small to moderate samples. This paper develops a rigorous framework for estimating Weibull parameters by combining the maximum-entropy principle with fractional-order moment constraints, i.e., constraints of the form E[Xr] for non-integer r>0. A central theme of the paper is symmetry: we show that the maximum-entropy density subject to a finite set of fractional-moment constraints uniquely recovers the Weibull family, and that the underlying moment-matching system, while not symmetric in every sense considered in an earlier draft (see Remark ), possesses a precisely characterised duality under joint rescaling and relabelling of the exponent pair, together with a log-moment map whose local curvature is strictly positive and increasing with exponent spacing rather than symmetric about a fixed midpoint. We derive closed-form expressions relating the Lagrange multipliers to the shape and scale parameters, establish new sound theoretical results on the symmetric behaviour of the moment-ratio function and its sensitivity, and propose a numerically stable algorithm for solving the resulting moment-matching system. Extensive Monte Carlo experiments demonstrate that the proposed maximum-entropy fractional-moment estimator achieves a bias and root-mean-square error that are comparable to, and for small-to-moderate samples somewhat better than, maximum likelihood estimation, with the size of the advantage depending on how closely the chosen exponent pair tracks the true shape parameter. Applications to real wind-speed data and composite-material fatigue life data illustrate the practical utility of the method. The paper contributes both to the information-theoretic foundations of distribution fitting and to applied statistical methodology, with symmetry serving as both a diagnostic tool and a unifying structural principle throughout.