A data transformation process for using Benford’s Law with bounded data

Benford’s Law is an empirical observation about the frequency of digits in a variety of naturally occurring data sets. Auditors and forensic scientists have used Benford’s Law to detect erroneous data in accounting and legal usage. One well-known limitation is that Benford’s Law fails when data have clear minimum and maximum values. Many kinds of education data, including assessment scores, typically include hard maximums and therefore do not meet the parametric assumptions of Benford’s Law. This paper implements a transformation procedure which allows for assessment data to be compared to Benford’s Law. As a case study, a data quality assessment of oral language scores from the Early Childhood Longitudinal Study, Kindergarten (ECLS-K) study is used and higher risk data segments detected. The same method could be used to evaluate other concerns, such as test fraud, or other bounded datasets.

An Alternative One-Parameter Distribution for Bounded Data Modeling Generated from the Lambert Transformation

The beta and Kumaraswamy distributions are two of the most widely used distributions for modeling bounded data. When the histogram of a certain dataset exhibits increasing or decreasing behavior, one-parameter distributions such as the power, Marshall–Olkin extended uniform and skew-uniform distributions become viable alternatives. In this article, we propose a new one-parameter distribution for modeling bounded data, the Lambert-uniform distribution. The proposal can be considered as a natural alternative to well known one-parameter distributions in the statistical literature and, in certain scenarios, a viable alternative even for the two-parameter beta and Kumaraswamy distributions. We show that the density function of the proposal tends to positive finite values at the ends of the support, a behavior that favors good performance in certain scenarios. The raw moments are derived from the moment-generating function and used to describe the skewness and kurtosis behavior. The quantile function is expressed in closed form in terms of the Lambert W function, which allows reparameterizing the distribution such that the involved parameter represents the qth quantile. Thus, for the analysis of a bounded range variable, for which a set of covariates is available, we propose a regression model that relates the qth quantile of the response to a linear predictor through a link function. The parameter estimation is carried out using the maximum likelihood method and the behavior of the estimators is evaluated through simulation experiments. Finally, three application examples are considered in order to illustrate the usefulness of the proposal.

Regulated seasonal unit root process

Unfortunately, time series problems do not appear in data singly. We focus on the joint occurrence of nonstationarity, seasonality and bounded data. Seasonal unit root tests and bounded unit root tests already exist in the literature, yet when all these issues are combined their performance needs improvement. That is why we offer a testing procedure for bounded seasonal unit root processes. The combination of these tests is not straightforward as the nonlinearity coming from bounds causes the limiting distribution of the proposed test statistic to be multivariate Brownian motion while the others have univariate distributions. The simulation exercises reveal that the existing tests, which ignores the presence of bounds or seasonality, suffer significant size problems. Our statistic removes the size distortions and also maintain satisfactory power performance.

A New Quantile Regression for Modeling Bounded Data under a Unit Birnbaum–Saunders Distribution with Applications in Medicine and Politics

Quantile regression provides a framework for modeling the relationship between a response variable and covariates using the quantile function. This work proposes a regression model for continuous variables bounded to the unit interval based on the unit Birnbaum–Saunders distribution as an alternative to the existing quantile regression models. By parameterizing the unit Birnbaum–Saunders distribution in terms of its quantile function allows us to model the effect of covariates across the entire response distribution, rather than only at the mean. Our proposal, especially useful for modeling quantiles using covariates, in general outperforms the other competing models available in the literature. These findings are supported by Monte Carlo simulations and applications using two real data sets. An R package, including parameter estimation, model checking as well as density, cumulative distribution, quantile and random number generating functions of the unit Birnbaum–Saunders distribution was developed and can be readily used to assess the suitability of our proposal.

The one-phase fractional Stefan problem

We study the existence and properties of solutions and free boundaries of the one-phase Stefan problem with fractional diffusion posed in [Formula: see text]. In terms of the enthalpy [Formula: see text], the evolution equation reads [Formula: see text], while the temperature is defined as [Formula: see text] for some constant [Formula: see text] called the latent heat, and [Formula: see text] stands for the fractional Laplacian with exponent [Formula: see text]. We prove the existence of a continuous and bounded selfsimilar solution of the form [Formula: see text] which exhibits a free boundary at the change-of-phase level [Formula: see text]. This level is located at the line (called the free boundary) [Formula: see text] for some [Formula: see text]. The construction is done in 1D, and its extension to [Formula: see text]-dimensional space is shown. We also provide well-posedness and basic properties of very weak solutions for general bounded data [Formula: see text] in several dimensions. The temperatures [Formula: see text] of these solutions are continuous functions that have finite speed of propagation, with possible free boundaries. We obtain estimates on the growth in time of the support of [Formula: see text] for solutions with compactly supported initial temperatures. Besides, we show the property of conservation of positivity for [Formula: see text] so that the support never recedes. On the contrary, the enthalpy [Formula: see text] has infinite speed of propagation and we obtain precise estimates on the tail. The limits [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are also explored, and we find interesting connections with well-studied diffusion problems. Finally, we propose convergent monotone finite-difference schemes and include numerical experiments aimed at illustrating some of the obtained theoretical results, as well as other interesting phenomena.