General flation models for count data

Abstract

The paper discusses very general extensions to existing inflation models for discrete random variables, allowing an arbitrary set of points in the sample space to be either inflated or deflated relative to a baseline distribution. The term flation is introduced to cover either inflation or deflation of counts. Examples include one-inflated count models where the baseline distribution is zero-truncated and count models for data with a few unusual large values. The main result is that inference about the baseline distribution can be based solely on the truncated distribution which arises when the entire set of flation points is truncated. A major application of this result relates to estimating the size of a hidden target population, and examples are provided to illustrate our findings.

Estimating Population Size of Criminals: A New Horvitz–Thompson Estimator under One-Inflated Positive Poisson–Lindley Model

“…A simple formula on the flation (inflation or deflation) parameter is given in Proposition 1 by Böhning and Ogden (2020).…”

“…where ∑ y = 1 ∞ n y = n . Böhning and Ogden (2020) provided inference on models with general flation, that is, inflation or deflation of counts with zero-truncated distribution act as a baseline distribution. By differentiating l with respect to ω and θ , and by setting the derivatives to 0, the maximum likelihood estimators for ω and θ are given respectively as…”

“…The latter equation in equation (4) can be solved numerically. Note that, the flation parameter used by Böhning and Ogden (2020) is compliment to the ω here. However, one can reach similar estimator for ω by taking one minus flation parameter estimator by Böhning and Ogden (2020).…”

“…The authors have also considered obtaining estimates by truncating singletons completely (Kaskasamkul & Böhning, 2018). When the truncated count data are inflated or deflated at any points in the count data, one can refer to the work of Böhning and Ogden (2020) for the statistical inference as well as for estimating the size of the hidden population.…”

Population size estimation based upon zero-truncated, one-inflated and sparse count data

“…One-inflation models have recently experienced some attention in capture-recapture modelling; see Böhning and van der Heijden (2019), Böhning et al (2019), Farcomeni (2020), Godwin (2017Godwin ( , 2019, or Godwin and Böhning (2017). In Böhning and Ogden (2020) a more general investigation of inflation models is delivered and their close connection to truncation models established. We see two major objectives for our paper:…”