Zero-inflated Bell regression models for count data

ABSTRACT

By starting from the one-parameter Bell distribution proposed recently in the statistic literature, we introduce the zero-inflated Bell family of distributions. Additionally, on the basis of the proposed zero-inflated distribution, a novel zero-inflated regression model is proposed, which is quite simple and may be an interesting alternative to usual zero-inflated regression models for count data. We consider a frequentist approach to perform inferences, and the maximum likelihood method is employed to estimate the zero-inflated Bell regression parameters. Monte Carlo simulations indicate that the maximum likelihood method is quite effective to estimate the zero-inflated Bell regression parameters. We also propose the Pearson residuals for the new zero-inflated regression model to assess departures from model assumptions. Additionally, the global and local influence methods are discussed. In particular, the normal curvature for studying local influence is derived under case weighting perturbation scheme. Finally, an application to the count of infected blood cells is considered to illustrate the usefulness of the zero-inflated Bell regression model in practice. The results suggest that the new zero-inflated Bell regression is more appropriate to model these count data than other familiar zero-inflated (or not) regression models commonly used in practice.

KEYWORDS:

• Bell distribution
• count data
• excess zeros
• overdispersion
• zero-inflated models

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 62F10
• 62J05
• 62J20

Acknowledgments

Germán Moreno–Arenas acknowledges grants from Mobility Program of the Universidad Industrial de Santander (Colombia). Fredy Castellares gratefully acknowledges the financial support from FAPEMIG (Belo Horizonte/MG, Brazil). We also thank two anonymous referees for constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. In this paper, all simulated envelopes were constructed by using K=2000, and $\gamma =1\mathrm{\%}$ thus having a 99% confidence band for the residuals.

Additional information

Funding

Artur Lemonte acknowledges the financial support of the Brazilian agency Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (grant 301808/2016–3).