A discrete version of the half-normal distribution and its generalization with applications

A new discrete distribution depending on two parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$\alpha >-1$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\sigma >0$$</EquationSource> </InlineEquation> is obtained by discretizing the generalized normal distribution proposed in García et al. (Comput Stat and Data Anal 54:2021–2034, <CitationRef CitationID="CR6">2010</CitationRef>), which was derived from the normal distribution by using the Marshall and Olkin (Biometrika 84(3):641–652, <CitationRef CitationID="CR16">1997</CitationRef>) scheme. The particular case <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\alpha =1$$</EquationSource> </InlineEquation> leads us to the discrete half-normal distribution which is different from the discrete half-normal distribution proposed previously in the statistical literature. This distribution is unimodal, overdispersed (the responses show a mean sample greater than the variance) and with an increasing failure rate. We revise its properties and the question of parameter estimation. Expected frequencies were calculated for two overdispersed and underdispersed (the responses show a variance greater than the mean) examples, and the distribution was found to provide a very satisfactory fit. Copyright Springer-Verlag Berlin Heidelberg 2014