On a simple unreferenced distribution with support [-1,1]

Year 2025, Volume: 1 Issue: 1, 1 - 25, 20.06.2025

Christophe Chesneau

Abstract

The notion of distributions is central to probability theory and statistics.
Because so many already exist, it is difficult to find new distributions that combine simplicity and originality. In this article, however, we present one, described as a simple but not yet referenced one-parameter distribution with support $[-1,1]$. It can be seen as a modified and restricted version of the logistic distribution on the interval $[-1,1]$, and also as a generalization of the uniform distribution over $[-1,1]$. In particular, the forms of the corresponding probability density functions are monotonic, typical of those obtained by a cumulative distribution function and its vertically symmetric version. We discuss the main properties of this new distribution, focusing on those related to the quantile function and the moments. Some additional results contribute to the theory of distributions, including one that introduces the notion of opposite distributions. The article concludes with a short numerical study, part of which is devoted to the maximum likelihood estimation of the unique parameter involved. It also includes the analysis of score-type data.

Keywords

probability, probability density function, logistic distribution, uniform distribution, statistical modeling, score-type data

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