Chow and Teugels (1978), followed by Anderson and Turkman (1991), have shown that under suitable conditions the sum and the maximum of a sequence of random variables are asymptotically independent. Examples -- both simple simulations and climatological applications -- exist which illustrates that the sum and the maximum of a moderately sized sample are positively correlated; that is, the asymptotic result has yet to be realized. To model this dependent structure, we derive a higher order expansion term for the joint density of the sum and the maximum of an iid sequence of random variables. We have indeed obtained uniform results for expansions of the joint density under all three domains of attraction for the maximum.

In the past 5 to 10 years, there has been a growing interest in climate change research on extreme climate events and their impact on the overall climate. With respect to U.S. precipitation, Karl and Knight (1998) claim that the increase in the extreme rainfalls is driving the increase in the total rainfall, indicating a change in the joint density. Using the higher order expansion which should provide a better methodology, we analyze continental U.S. precipitation from 1901 to 1997, focusing on the trends in the annual total and annual maximum rainfall. We highlight the results which show the impact the higher order expansion has had on the analysis. In doing so, we conclude that not only are both trends increasing but also that the shift in the mean does not completely account for the change in the extremes.