By Rainer Winkelmann

The publication presents an up to date survey of statistical and econometric concepts for the research of count number info, with a spotlight on conditional distribution types. The booklet starts off with a presentation of the benchmark Poisson regression version. replacement versions tackle unobserved heterogeneity, kingdom dependence, selectivity, endogeneity, underreporting, and clustered sampling. trying out and estimation is mentioned. ultimately, purposes are reviewed in numerous fields.

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**Example text**

The resulting estimator won't have the standard properties of a maximum likelihood estimator. Alternatively, one can treat n as a continuous parameter. In this case, derivatives can be taken. Since n! (n-k)! r(n + 1) r(k+1)r(n-k+1) where r(·) denotes the gamma-function and r(n + 1) = n! if n is an integer, this involves computation of the digamma function. Alternatively, direct differentiation can be based on an approximation of the factorial representation using Stirling's formula k! ~ (21T)1/2kk+1/2 exp (_k){1 + 1/12k} In either case, a logical difficulty arises with respect to the possible sample space of the underlying random variable X if n is a continuous non-negative parameter.

The use of this model in regression analysis requires an expression for the mean. 87) The approximation is exact for c = 0 and for c ---* 1. L + b)1-c 1-c bl - c ~--~--------- This generalized count data model has two more parameters than the Poisson model. L can be expressed in terms of covariates, and the parameters of the model can be estimated by maximum likelihood. 4 Katz Family of Distributions Distributions for non-negative integers can be uniquely represented by their recursive probability ratios [(X P X = ~ k) ) = ~ = f(k, B) - 1 Pk-l k = 1,2, ...

Example 3 Let N be the number of spells of illness and X be the number of visits to a doctor in a given spell. Z gives then the total number of visits to a doctor. Example 4 Let X be a degenerate random variable such that P(X = 1) = 1. Then Z = N. The concept of compounding is not restricted to cases where N is a count random variable although this is of main interest here. One popular count data model, the hurdle Poisson model, can be interpreted as a Bernoulli distribution that is generalized by a truncated-at-zero Poisson distribution (Santos Silva and Windmeijer, 1999).