On Asymptotic Simultaneous Confidence Regions for Regression Planes in a Poisson Model

Abstract

analysis it is then necessary to use a model, which accounts for the effect of these variables. It is sometimes reasonable to assume that the external variables have an influence only on the means of the random variables studied, and that a regression model is appropriate. Let us, to fix ideas, think of the following situation. In the study of road safety one important question is how to judge the effect of certain imposed measures or changes in the traffic milieu. In order to judge for example the effect on the accident development of an imposed speed limit, on a certain road net during a certain time period, it seems reasonable to take such things as type and amount of traffic, weather and road conditions into account. One possible approach is then to assume the expected number of accidents to depend on such explaining factors, in other words to assume a regression model. By comparing the corresponding means one can then get an idea of the accident development on different road nets or during different time periods. It is in this connection of course essential to be able to state the precision of the estimated means, to make confidence statements. Since in the situation referred to one is interested in comparing the accident development between say two time periods, one also wants to make confidence statements for the difference between two means or regression planes. We will in the application part of this paper give some examples of confidence regions for the expected number of accidents, as well as for the difference between two such means, in the case the expected number of accidents a certain day is assumed to be a linear function of the estimated number of vehicle kilometers travelled that day. In several cases, when one wishes to apply a regression model to a set of observations it is known that the dependent variable is discrete and positive. Then it may sometimes be more appropriate to use a regression model where the dependent variable is assumed to be Poisson distributed than to use a normal regression model. However, as compared with the case in which one has assumed a normal distribution, the assumption of a Poisson distribution implies some complications. To begin with the parameter, the mean, of a Poisson distribution is positive. This means that for given values of the regressors, the possible values of the regression coefficients are