By Alfred Lodge, Oliver Lodge

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Differential calculus for beginners (1905)

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Number at risk), and x the known set of k explanatory variables. The mean number of events μ may be expressed as the product of N and π, the probability of the occurrence of event, sometimes also called the hazard rate. That is, μ(x) = N(x)π(x, β). In this case the probability π is parameterized in terms of covariates. For example, π = exp(x β). 3 Examples 11 to a rate form of the Poisson model with the density Pr [Y = y|N (x), x] = e−μ(x) μ(x)y , y! y = 0, 1, 2, . . 19) Variants of the Poisson regression arise in a number of ways.

Hence, Chapter 8 deals more with a number of special cases and provides relatively few results of general applicability. Another class of multivariate models uses longitudinal or panel data, which are analyzed in Chapter 9. Longitudinal count models have attracted much attention, following the work of Hausman, Hall, and Griliches (1984). Such models are relevant if the regression analysis is based on (yit , xit ), i = 1, . . , n, t = 1, . . , T , where i and t are individual and time subscripts, respectively.

Often the number of defaulted payments is studied as a count variable. Greene (1994) analyzes the number of major derogatory reports made after a delinquency of 60 days or longer on a credit account, for 1,319 individual applicants for a major credit card. Major derogatory reports are found to decrease with increases in the expenditure-income ratio (average monthly expenditure divided by yearly income). Age, income, average monthly credit-card expenditures, and whether the individual holds another credit card are statistically insignificant.

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