Abstract
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.
Highlights
It is common to deal with data expressed as a proportion, percentage, rate or fraction in the continuous range (0, 1) when analyzing certain random phenomena, for example, when observing the annual replacement rate related to blue collar workers [1], the proportion of codling moth eggs that die from fumigation with methyl bromide [2] and the percentage difference in nicotine levels in users of first and new generation e-cigarette devices [3]
Two widely used probability distributions in data modeling such as those described above are the two-parameter beta (B) [4] and Kumaraswamy (K) [5] distributions. These distributions have a very flexible probability density function, presenting monotonic, unimodal and U shapes. These distributions are usually the first alternatives considered for modeling bounded data, it is possible to find in the statistical literature one-parameter distributions that can appropriately model datasets whose histograms show increasing or decreasing behavior
The power (P) distribution, which can be derived as a special case of the B and K distributions, and the Marshall–Olkin extended uniform (MOEU) [6] and skew-uniform (SU) [7] distributions, which are the result of the popular approach of adding a parameter to a baseline distribution in search of a more flexible distribution, can be considered viable alternatives
Summary
It is common to deal with data expressed as a proportion, percentage, rate or fraction in the continuous range (0, 1) when analyzing certain random phenomena, for example, when observing the annual replacement rate related to blue collar workers [1], the proportion of codling moth eggs that die from fumigation with methyl bromide [2] and the percentage difference in nicotine levels in users of first and new generation e-cigarette devices [3]. These distributions have a very flexible probability density function (pdf), presenting monotonic, unimodal and U shapes These distributions are usually the first alternatives considered for modeling bounded data, it is possible to find in the statistical literature one-parameter distributions that can appropriately model datasets whose histograms show increasing or decreasing behavior. We formulate the following question as a starting point in this work: Based on the approach of adding parameters to a baseline distribution, is it possible to generate a parsimonious distribution that can perform better than the P, MOEU, SU, B and K distributions when modeling data whose histogram exhibits increasing or decreasing behavior?
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