An AI tool’s effectiveness relies upon the degree to which its training data possesses cluster points. The riskiness of an assets projected return is inversely related to the peakedness of its returns distribution. Distributions within a mixture are more easily identified the more concentrated they are around their centrality points and the less they overlap. All are matters of informity which, at the basic distributional level, relates to the extent to which an outcome distribution is peaked and not uniformly distributed. While Pearsons Kurtosis Measure has been the conventional gauge of Peakedness, it has been demonstrated to be more about tail thickness than modal peakedness. Here generic Peakedness or Informity measures, applicable in any multidimensional discrete or continuous data environment are developed. The measures have an upper bound of 1 representing maximal peakedness or informity and a lower bound of 0 representing absence of a peak or complete lack of informity. In addition, because of the association of a distribution’s peakedness with its modality, modally focused Kurtosis and Coefficient of Variation measures are proposed for use in asymmetrical distributional contexts where mean oriented measures may not be appropriate. The new measures are exemplified in a variety of data scenarios.
The Gini index is an important measure of income inequality, and calculating it under different conditions is highly significant. Left truncated and right censored data are common in economics, while existing literature on Gini index estimation often focuses on right censoring. We construct a nonparametric estimator for the Gini index when data are subject to random left truncated and right censored, and investigate its consistency and asymptotic normality. Simulations and a real data example demonstrate that it has certain advantages when compared to existing methods.
To achieve greater flexibility for modelling heavy-tailed responses on the unit interval, a beta scale mixture (BSM) regression model is proposed. The conditional response is assigned a mean-parameterised beta distribution whose variability parameter is scaled by a mixing random variable, taking values on all or part of the positive real line, whose distribution depends on parameters governing the tail behaviour of the resulting compound distribution. The conditional mean of the response variable is linked to covariates through a logit link, while the mixing mechanism offers greater flexibility towards the skewness and kurtosis than classical beta regression. To validate the effectiveness of the proposed regression model, we conduct a simulation study and illustrate its practical relevance through applications to real datasets. The results indicate that the BSM regression model consistently outperforms the classical beta and alternative competing regression models for responses on the bounded unit domain. The methodology is implemented in the open-source BSMreg package for R, available at https://github.com/arnootto/BSM .
Incomplete data, such as missing or censored values, frequently arise in matrix-variate datasets. At the same time, latent heterogeneity–manifested as subpopulations or clusters–is often present. To our knowledge, no finite mixture model has yet been proposed that can simultaneously handle matrix-variate data with both missing and censored values. To address this gap, we introduce a novel model based on the matrix-variate normal distribution that accommodates missingness and various censoring types while capturing latent group structure through clustering. The missing/censoring mechanism is assumed to be missing at random, implying a non-informative (ignorable) mechanism. An efficient expectation-conditional maximization algorithm is developed for maximum likelihood estimation. Simulations demonstrate accurate parameter recovery under a variety of censoring and missingness patterns. Applications of the method to environmental data from the Chesapeake Bay and clinical data from the Age-Related Eye Disease Study illustrate its ability to reveal ecologically and clinically meaningful structures in complex, incomplete datasets.
This paper describes an interesting class of discrete probability distributions that arise from the discretization of continuous distributions. They are sometimes also called telescopic distributions, a term inspired by the concept of “telescoping series" in mathematics. After some introductory remarks, we will focus primarily on deriving two-dimensional telescopic distributions, i.e., discretized two-dimensional continuous probability distributions, which are typically constructed using a suitable copula. We will then present examples using both simulated and real data to demonstrate the advantages and potential applications of these models. Among other things, we will derive and apply the “telescopic" form of the bivariate negative binomial distribution.
