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Income modeling is crucial in determining workers’ earnings and is an important research topic in labor economics. Traditional regressions based on normal distributions are statistical models widely applied. However, income data have an asymmetric behavior and are best modeled by non-normal distributions. The objective of this work is to propose parametric quantile regressions based on two asymmetric income distributions: Dagum and Singh–Maddala. The proposed quantile regression models are based on reparameterizations of the original distributions by inserting a quantile parameter. We present the reparameterizations, properties of the distributions, and the quantile regression models with their inferential aspects. We proceed with Monte Carlo simulation studies, considering the performance evaluation of the maximum likelihood estimation and an analysis of the empirical distribution of two types of residuals. The Monte Carlo results show that both models meet the expected outcomes. We apply the proposed quantile regression models to a household income data set provided by the National Institute of Statistics of Chile. We show that both proposed models have good performance in model fitting. Thus, we conclude that the obtained results favor the Singh–Maddala and Dagum quantile regression models for positive asymmetrically distributed data related to incomes. The economic implications of our investigation are discussed in the final section. Hence, our proposal can be a valuable addition to the tool-kit of applied statisticians and econometricians.

A fractile is a location on a probability density function with the associated surface being a proportion of such a density function. The present study introduces a novel methodological approach to modeling data within the continuous unit interval using fractile or quantile regression. This approach has a unique advantage as it allows for a direct interpretation of the response variable in relation to the explanatory variables. The new approach provides robustness against outliers and permits heteroscedasticity to be modeled, making it a tool for analyzing datasets with diverse characteristics. Importantly, our approach does not require assumptions about the distribution of the response variable, offering increased flexibility and applicability across a variety of scenarios. Furthermore, the approach addresses and mitigates criticisms and limitations inherent to existing methodologies, thereby giving an improved framework for data modeling in the unit interval. We validate the effectiveness of the introduced approach with two empirical applications, which highlight its practical utility and superior performance in real-world data settings.

The need for comparing two regression functions arises frequently in statistical applications. Comparison of the usual regression functions is not very meaningful in situations where the distributions and the ranges of the covariates are different for the populations. For instance, in econometric studies, the prices of commodities and people's incomes observed at different time points may not be on comparable scales due to inflation and other economic factors. In this article, we describe a method of standardizing the covariates and estimating the transformed regression function, which then become comparable. We develop smooth estimates of the fractile regression function and study its statistical properties analytically as well as numerically. We also provide a few real examples that illustrate the difficulty in comparing the usual regression functions and motivate the need for the fractile transformation. Our analysis of the real examples leads to new and useful statistical conclusions that are missed by comparison of the usual regression functions.

In the evolving landscape of psycholinguistic research, this study addresses the inherent complexities of data through advanced analytical methodologies, including permutation tests, bootstrap confidence intervals, and fractile or quantile regression. The methodology and philosophy of our approach deeply resonate with fractal and fractional concepts. Responding to the skewed distributions of data, which are observed in metrics such as reading times, time-to-response, and time-to-submit, our analysis highlights the nuanced interplay between time-to-response and variables like lists, conditions, and plausibility. A particular focus is placed on the implausible sentence response times, showcasing the precision of our chosen methods. The study underscores the profound influence of individual variability, advocating for meticulous analytical rigor in handling intricate and complex datasets. Drawing inspiration from fractal and fractional mathematics, our findings emphasize the broader potential of sophisticated mathematical tools in contemporary research, setting a benchmark for future investigations in psycholinguistics and related disciplines.

Fractile Graphical Analysis (FGA) was proposed by Prasanta Chandra Mahalanobis in 1961 as a method for comparing two distributions at two different points (of time or space) controlling for the rank of a covariate through fractile groups. We use bootstrap techniques to formalize the heuristic method used by Mahalanobis for approximating the standard error of the dependent variable using fractile graphs from two independently selected \"interpenetrating network of subsamples.\" We highlight the potential and revisit this underutilized technique of FGA with a historical perspective. We explore a new non-parametric regression method called Fractile Regression where we condition on the ranks of the covariate and compare it with existing regression techniques. We apply this method to compare mutual fund inflow distributions after conditioning on ranks or fractiles of pre-tax and post-tax returns and compare distributions of private and public equity returns after controlling for fractiles of assets under management size using the two sample smooth test.