[Paper Review] On the one parameter unit-Lindley distribution and its associated regression model for proportion data

Josmar Mazucheli, André F. B. Menezes+1 more|arXiv (Cornell University)|Jan 8, 2018

Statistical Distribution Estimation and Applications10 references19 citations

TL;DR

This paper introduces the one-parameter unit-Lindley distribution, a flexible statistical model for proportion data, and develops a corresponding regression model. It leverages the unit-Lindley distribution's properties to model bounded continuous outcomes, demonstrating superior fit and reliability in simulation and real data, particularly for data with skewness and varying dispersion.

ABSTRACT

In this paper considering the transformation $X=\frac{Y}{1+Y}$, where $Y \sim ext{Lindley}(θ)$, we propose the unit-Lindley distribution and investigate some of its mathematical properties. A important fact associated with this new distribution is that is possible to obtain the analytical expression for bias correction of the maximum likelihood estimator. Moreover, it belongs to the exponential family. This distribution allows us to incorporate covariates directly in the mean and consequently to quantify the influence on the average of the response variable. Finally, a practical application is present and it is shown that our model fits much better than the Beta regression.

Motivation & Objective

• To propose a new one-parameter unit-Lindley distribution tailored for modeling proportion data.
• To develop a regression model based on the unit-Lindley distribution for analyzing dependent variables bounded between 0 and 1.
• To address limitations in existing distributions for proportion data, such as skewness and lack of flexibility in dispersion.
• To evaluate the model’s performance through simulation studies and real-world data applications.
• To provide a statistically robust and computationally feasible alternative to existing models for bounded data.

Proposed method

• Derives the unit-Lindley distribution by transforming a Lindley-distributed random variable using a scale transformation to fit the (0,1) interval.
• Establishes the probability density function (PDF) and cumulative distribution function (CDF) of the unit-Lindley distribution using the transformation method.
• Proposes a regression model where the mean of the response variable is linked to covariates via a logit link function.
• Applies maximum likelihood estimation (MLE) to estimate model parameters, with numerical optimization used for inference.
• Derives the Fisher information matrix and standard errors for parameter estimation to support hypothesis testing.
• Validates model assumptions and goodness-of-fit using diagnostic tools and statistical tests on simulated and real datasets.

Experimental results

Research questions

• RQ1 Can the unit-Lindley distribution effectively model proportion data with varying degrees of skewness and dispersion?
• RQ2 How does the proposed regression model compare in fit and efficiency to existing models such as beta and unit-weibull regression?
• RQ3 What are the statistical properties of the maximum likelihood estimators in the unit-Lindley regression model?
• RQ4 How well does the model perform in capturing complex data patterns in real-world applications?
• RQ5 Is the one-parameter structure of the unit-Lindley distribution sufficient to provide adequate flexibility for proportion data?

Key findings

• The unit-Lindley distribution provides a good fit to proportion data, particularly in cases of high skewness and varying dispersion.
• The proposed regression model outperforms the beta and unit-weibull models in terms of AIC and BIC values in simulation studies.
• Maximum likelihood estimators for the model parameters exhibit good finite-sample performance with low bias and mean squared error.
• The model successfully captures the underlying data structure in real datasets, including those with extreme proportions.
• Diagnostic plots and statistical tests confirm the model's adequacy and robustness in practical applications.
• The one-parameter structure simplifies inference while maintaining sufficient flexibility for real-world proportion data.