By Rainer Winkelmann

This ebook provides statistical equipment for the research of occasions. the first concentration is on unmarried equation move part versions. The ebook addresses either the technique and the perform of the topic and it offers either a synthesis of a various physique of literature that hitherto used to be to be had principally in items, in addition to a contribution to the growth of the technique, setting up numerous new effects and introducing new types. ranging from the traditional Poisson regression version as a benchmark, the motives, indicators and results of misspecification are labored out. either parametric and semi-parametric choices are mentioned. whereas semi-parametric types let for powerful interference, parametric versions can determine beneficial properties of the underlying information iteration strategy.

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Clearly, positive duration dependence negatively affects the probability of long spells, while negative duration dependence does the opposite. No duration dependence corresponds to the case of a constant hazard. The hazard function of an interarrival time distribution does not need to be monotonic although many parametric distributions that are used commonly have monotonic hazard functions. J(t) =- dt logF(t). 69) Thus, F(t) = 1 - e-~t and f(t) = Ae-~t. g. DeGroot, 1986). The hazard (which equals the inverse of the expected value) is constant if and only if the distribution of completed spells is exponential.

G. DeGroot, 1986). 26) is parameterized in terms of iJ = 1/0. If o: is an integer, the distribution is also called Pascal distribution. Another special case is the geometric distribution which is a negative binomial distribution with o: = 1. 3. Distributions for Count Data 31 For the later use in econometric applications an alternative formulation avoiding the r-terms will prove more convenient for computational purposes. -"- j=l 1 +0 J fork= 1, 2, ... 30) Since () Var(X) = a0(1 + 0) . > 0, the variance of the negative binomial distribution always exceeds its mean ('overdispersion').

But the equality holds if and only if a= 0 and (b = 0 orb= 1). Thus, Y cannot have a Poisson distribution. 0 Chapter 2. Probability Models for Event Counts 28 In fact, it can be shown that within a large class of distributions, the normal distribution is the only distribution preserved under both location and scale transformations (see Hinkley and Reid, 1991). 2. 22) = 0, 1, ... ,n. 23) and mean and variance are E(X) = np and Var(X) = np(1 - p), respectively. In estimation problems, the binomial parameter n is usually treated as given.