Robust parameter estimation for one-inflated positive Poisson Lindley distribution under the presence and absence of outliers with applications to crime data

Abstract

The one-inflated positive Poisson Lindley model has been recently introduced as an alternative in modelling positive count data with a large number of ones: a phenomenon known as one-inflation. In the presence of one-inflation, this model has a high tendency to be influenced by outliers, making usual parameter estimations to be less robust. Hence, several estimators: maximum likelihood, method of moments, ordinary least squares, weighted least squares, Cramér-Von Mises, modified Cramér-Von Mises (MCVM) and maximum product of spacing (MPS); for the parameters of the model are also proposed and investigated in terms of unbiasedness, consistency and joint efficiency under the presence and absence of outliers. When the outliers are absent, the MPS estimator is the best estimator and when the outliers are present, the MCVM estimator is the best estimator. Model fittings to two real datasets with one-inflation and outliers support the simulation results and conclude that the MCVM estimator is the best estimator. Based on the best robust estimator, the population size of the number of offenders as well as the likelihood of arrests were estimated.