Monometrics induced by quasi-arithmetic mean operators

The beta kernel estimator offers a theoretically superior alternative to the Gaussian kernel for unit interval data, eliminating boundary bias without requiring reflection or transformation. However, its adoption remains limited by the lack of a reliable bandwidth selector, and practitioners currently rely on computationally expensive iterative optimization methods that are prone to instability. We derive the “Beta Reference Rule”, a fast, closed-form bandwidth selector, based on the unweighted asymptotic mean integrated squared error (AMISE) of a beta reference distribution. To address boundary integrability issues, we introduce a principled heuristic for U-shaped and J-shaped distributions. By employing a method-of-moments approximation, we reduce the bandwidth selection complexity from iterative optimization to O ( 1 ) . Extensive Monte Carlo simulations demonstrate that our rule matches the accuracy of numerical optimization while delivering a speedup of over 35,000 times. Real-world validation on socioeconomic data shows that it avoids the “vanishing boundary” and “shoulder” artifacts common to Gaussian-based methods. We provide a comprehensive, open-source Python package to facilitate the immediate adoption of the beta kernel as a drop-in replacement for standard density estimation tools.

Abstract This paper introduces a novel framework for understanding intuitionistic values and fuzzy sets through the automorphism of the unit interval [0, 1]. By generalizing the concepts of classical intuitionistic sets using strong negations generated by these automorphisms, we define the φ -intuitionistic values and explore their arithmetic, establishing a comprehensive set of operations based mainly on additive generators of t-(co)norms and their properties. This approach provides greater flexibility to the modelling with these sets.

ABSTRACT This paper addresses the gap in subnational assessments of Sustainable Development Goals (SDGs) by benchmarking the performance of European Union NUTS‐2 regions in the People and Prosperity pillars of the 2030 Agenda for Sustainable Development. We construct composite indices using a methodological framework that combines the “benefit‐of‐the‐doubt” approach with cross‐efficiency Data Envelopment Analysis, enabling an assessment of both overall performance and the balance across SDG indicators. The resulting indices, which take values within the standard unit interval, are then regressed on a set of explanatory variables using beta regression models. The analysis uncovers substantial regional disparities and identifies positive associations between regional governance quality, innovation intensity, and balanced sustainable development outcomes. These findings provide valuable insights into the dynamics of regional sustainability within the European Union.

This paper presents a new distribution on the unit interval, named the Unit Arcsine-Rayleigh distribution (UASRD), which is the result of the exponential transformation of the Arcsine-Rayleigh distribution. The model suggested is versatile and can be used in modeling limited reliability and proportion data. Entropy-based measures are also studied to determine the uncertainty and information content of the proposed model and further explain the probabilistic nature of the proposed model and its potential applicability in information-theoretic and reliability tasks. These findings demonstrate the utility of the suggested model in the study of the limited data in the context of information theory. Basic statistical characteristics are derived, such as cumulative and density functions, quantile function, reliability and hazard functions, and ordinary moments. Estimation of parameters is obtained through approaches of maximum likelihood and maximum product spacing and Bayesian estimation of parameters. The performance of the estimators is also assessed by a Monte Carlo simulation study, and the application of real data shows the utility of the proposed model to the analysis of bounded data.

We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new $e$-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an $e$-positive graph. We also prove $e$-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes $e$-positivity of cycle-chord graphs and supports Ellzey's conjectured $e$-positivity for proper circular arc digraphs.

ABSTRACT We propose a new time series model for continuous data supported on the open unit interval , motivated by applications in environmental and energy systems. The Matsuoka autoregressive moving average (MARMA) model combines the Matsuoka distribution‐a uniparametric member of the canonical exponential family‐as the conditional distribution with a flexible ARMA‐type structure for the conditional mean. Parameters are estimated via partial maximum likelihood, allowing for random, time‐dependent covariates and enabling standard asymptotic inference. To construct out‐of‐sample prediction intervals, we explore a bootstrap‐based procedure that captures the uncertainty in the dynamic structure. A simulation study evaluates the finite‐sample performance of the method. The model is applied to the monthly proportion of electricity generated in the United States from all sources, except conventional hydropower. This application highlights the model's utility in capturing serial dependence, ensuring predictions remain within bounds, and providing reliable forecast intervals‐key features for robust energy system planning and environmental policy analysis.

In this paper, a new probability model is suggested, known as the Unit Linear Failure Rate Distribution (ULFRD), which is used to analyse data expressed on a unit interval (0, 1), e.g., proportions, rates, and normalised indices. The proposed model is a transformation of the classical linear failure rate distribution to finite domains and gives us the opportunity to have shapes with a variety of shapes that can model any hazard rate behaviour, such as bathtub-shaped ones that are common in reliability research. Various fundamental statistical features of the distribution are obtained. The parameter estimation is analysed under Type-II censoring, where maximum likelihood and Bayesian estimations are used. Bayesian estimates are obtained under a symmetric and an asymmetric loss of a Metropolis–Hastings within a Gibbs approximation. The analyses of the estimates’ performance are performed via a simulation study of various sample sizes and censoring plans. Lastly, the generalisability of the proposed model is also demonstrated with two real datasets in the socioeconomic and reliability settings. The findings prove that the ULFRD offers a flexible and competitive alternative to model-bound data.

This paper introduces the Unit Exponential Delay Time Distribution (UEDTD), a two-parameter model for data with support in the unit interval (0,1). The model is derived using two distinct approaches: transformation method applied to the Exponential Delay Time Distribution (EDTD), which itself arises as the convolution of two independent exponential random variables, and product convolution method of two independent power-function random variables that connects UEDTD to Pareto distribution, offering additional interpretability and giving rise to several exact and efficient algorithms for generating random samples. The limit distribution is examined with derivation of key statistical properties. The order statistics with interesting asymptotic results for extremes distribution are discussed and formulated. A reparameterization for the model is suggested to improve estimation stability and formulation with maximum likelihood approach employed for parameter inference. A simulation study demonstrates the consistency and efficiency of the estimators across various sample sizes and parameter configurations. The practical applicability of the UEDTD is demonstrated through a real-world dataset, where it shows superior performance compared to established unit distributions, confirming the utility of the UEDTD for modeling proportional data in applied research.

This paper presents a new continuous data model, the Unit Arcsine–Exponential distribution (UASED), a flexible data model on the unit interval. It is built up by an exponential-based arcsine-type transformation to allow it to represent a very wide range of shapes that can be used to model proportions and rates. A number of basic properties are obtained, such as closed-form formulas of the quantile function, moments, and entropy measures. Maximum likelihood and maximum product of spacings methods are developed to estimate parameters, and their performance is determined by Monte Carlo simulation, which shows that these methods can reasonably estimate the parameters and be stable over a variety of different parameter settings. To demonstrate that a model is practically useful, an application to real-world data on the reliability of devices in terms of failure time is discussed. The findings indicate that the UASED is a good fit to the data, in the sense that it is effective in terms of skewness and tail behavior and compares well or competes favorably with current unit distributions. All in all, the suggested model is a sparse alternative to model bounded data with sound inferential characteristics and high practical utility.

ABSTRACT In this study, we extend the additively generated triangular norms from the framework of the unit interval to that of partially ordered sets. We present several conditions under which this formula yields a t‐norm, where are partially ordered sets, and are monotone functions and is a t‐norm/t‐conorm on . The partially ordered semigroups induced by are order‐preserving/order‐reversing homomorphic to semigroup deformations of the semigroup induced by .

We prove necessary conditions for certain elementary symmetric functions, e λ , to appear with nonzero coefficient in Stanley’s chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs, giving more insight into the Stanley–Stembridge Conjecture, recently proven by Hikita, and the Shareshian–Wachs Conjecture. The Schur basis permits us to give a new interpretation of the coefficient of e n in terms of tableaux. We are also able to give an explicit formula for that coefficient.

This work establishes bounds for certain matrices that arise in the study of the convergence of expansions in Walsh functions, or Walsh–Fourier series. The matrices in question arise by “truncating” certain orthogonal matrices corresponding to expansions of dyadic step functions on the unit interval in the basis of Walsh functions. Here, truncation means, for each column, replacing all entries in that column below a column-dependent row by zero. The truncations correspond to partial sum operators in the Walsh basis. We study here a specific family of these truncated matrices that are shown elsewhere to have optimal norms among certain families of truncations. The main result here provides an approximate eigenvalue bound from which one can conclude that the norm of the truncation approaches a fixed value as the dimension of the truncation matrix approaches infinity. Its proof relies on the interplay between continuous and discrete sets. In particular, it is shown that integer samples of certain sinusoidal functions form approximate eigenvectors of a compressed version of the truncation. This bound plays an important role in a bigger new approach to the convergence of Walsh–Fourier series that this work is part of.

Abstract This paper introduces a new time series model based on the reflected unit Burr XII (RUBXII) distribution that is an alternative to the Kumaraswamy autoregressive moving average and Beta autoregressive moving average models for time series analysis taking values in the standard unit interval. The proposed model describes the conditional median of RUBXII-distributed discrete-time series by a dynamic structure that includes autoregressive and moving average (ARMA) terms, a set of regressors, and a link function. We perform the model’s parameter estimation using the conditional maximum likelihood method. Closed-form expressions for the score vector and observed information matrix are presented. We propose and discuss techniques of diagnostic and forecasting for the new model. A Monte Carlo simulation study is carried out to evaluate the finite sample performance of the conditional maximum likelihood estimator. Finally, the proportion of stored hydroelectric energy in Northern Brazil is analyzed through the proposed model. The results evidence that the introduced RUBXII-ARMA model is suitable for describing the dynamics of the data and provides more accurate forecasts for the proportion of stored energy in Northern Brazil than those from competitors’ models.

We study the additive and fractal structure of digit-restricted subsets of the unit interval where AD=∑n=1∞anb−n:an∈D⊆{0,…,b−1},|D|≥2, defined by allowing only digits from D in base-b expansions. These sets generalize the middle-third Cantor set and include a wide range of missing-digit and structured-digit fractals. We develop a rigorous framework for base-b digit arithmetic that separates purely discrete digit-combinatorics from carry effects. We give sharp sufficient criteria for intervals in AD+AD and AD−AD via carry-free digit blocks, establish an arithmetic obstruction to interval formation for all iterated sumsets, and prove a dimension-jump dichotomy: either a gcd obstruction prevents intervals for every k, or else some iterated sumset AD(k) contains an interval, and hence has full Hausdorff dimension 1. We also discuss the similarity-dimension formula under the open set condition, include definitions and preliminaries for a broad audience, and situate the results within classical and modern literature on Cantor sets and sumsets of self-similar sets.

ABSTRACT This paper introduces a novel unit‐Lindley mixed‐effects model (NULMM) within the generalized linear mixed model (GLMM) framework, designed for analyzing correlated response variables bounded within the unit interval. Parameter estimation was conducted via maximum likelihood, using Laplace approximation and adaptive Gaussian‐ Hermite quadrature (AGHQ). Simulation studies revealed that the Laplace approximation yielded biased estimates, while AGHQ with 5 or 11 quadrature points produced unbiased results. The proposed model was applied to rural electricity access data from South Asian countries, with covariates including time, log(GDP), log(Rural Population), and income level. Results show that time and log(GDP) are positively associated with rural electricity access, whereas log(Rural Population) has a negative association but is not statistically significant. Additionally, significant disparities were observed between low‐income and upper‐middle‐income countries. Model comparisons demonstrated that NULMM provides a better fit to the data than the beta mixed model and the unit‐Lindley (UL) mixed model.

In this article, we examine how income distribution and energy use shaped per-capita CO2 emissions between 1965 and 2022 at three spatial scales—world, northern hemisphere, and southern hemisphere. After re-scaling all the variables to the unit interval, we first estimate separate ordinary least squares regressions and then re-estimate the three equations jointly with Zellner’s seemingly unrelated regression, a step warranted by the substantial contemporaneous error correlation. Across both emission channels—aggregate CO2 and the land-use component—energy consumption emerges as the most consistent and statistically powerful predictor. Inequality effects are heterogeneous. The Gini coefficient amplifies emissions in the southern hemisphere and in the global system but is negligible in the northern fossil-fuel equation, while the Palma ratio reduces land-use emissions once overall inequality is held constant. Temperature anomalies display a further asymmetry, reducing land-use emissions everywhere, yet coinciding with higher fossil-fuel emissions in the south. Joint estimation raises the system-wide goodness of fit to 0.97 for land-use emissions and 0.99 for total emissions and yields more precise coefficients than the separate regressions. The results indicate that decarbonizing energy systems is a universal mitigation priority, whereas distributional reforms and land-governance measures are likely to deliver the greatest additional benefits in the southern hemisphere.

In this article, we extend the continuous Bernoulli to a shape-flexible distribution on the unit interval ( 0 , 1 ) . Unlike the well-known beta distribution, it admits a closed-form cumulative distribution function (CDF). The extended model preserves the analytical tractability of the original continuous Bernoulli distribution while allowing greater flexibility. We derive explicit expressions for the moments and, most notably, for the stress–strength reliability function in closed form. Maximum likelihood estimation is implemented via efficient fixed-point and scoring-based routines. We then develop a logistic link regression that treats stress–strength reliability as a covariate-dependent quality index for bounded outcomes, yielding interpretable reliability curves. An illustrative application demonstrates that the model captures complex reliability patterns and offers practical utility for reliability and quality engineering.

ON TWO NEW THEOREMS ON CONVEX INTEGRAL INEQUALITIES

We state combinatorial formulas for hyperoctahedral group ( 𝔅 n ) character evaluations of the form χ ( C ˜ w BC ( 1 ) ) where C ˜ w BC ( 1 ) ∈ ℤ [ 𝔅 n ] is a type- BC Kazhdan–Lusztig basis element, with w ∈ 𝔅 n corresponding to simultaneously smooth type- B and C Schubert varieties. We also extend the definition of symmetric group codominance to elements of 𝔅 n and show that for each element w ∈ 𝔅 n as above, there exists a BC -codominant element v ∈ 𝔅 n satisfying χ ( C ˜ w BC ( 1 ) ) = χ ( C ˜ v BC ( 1 ) ) for all 𝔅 n -characters χ . Combinatorial structures and maps appearing in these formulas are type- BC extensions of planar networks, unit interval orders, indifference graphs, poset tableaux, and colorings. Using the ring of type- BC symmetric functions, we introduce natural generating functions Y ( C ˜ w BC ( 1 ) ) for the above evaluations. These provide a new type- BC analog of Stanley’s chromatic symmetric functions [Adv. Math. 111 (1995) pp. 166–194].