Both intensities of individual extreme rainfall events and the frequency of such events are important for infrastructure planning. We develop a new statistical extreme value model, the PGEV model, which makes it possible to use high quality annual maximum series data instead of lesswell checked daily data to estimate trends in intensity and frequency separately. The method is applied to annual maxima data from the NOAA Atlas 14, Volume 10, dating from approximately 1900 to 2014, showing that in the majority of 333 rain gauge stations in the Northeastern USA the frequency of extreme rainfall events increases as mean temperature increases, but that there is little evidence of trends in the distribution of the intensities of individual extreme rainfall events. The median of the frequency trends corresponds to extreme rainfalls becoming 83% more frequent for each centigrade degree of temperature increase. Naturally, increasing trends in frequency also increase the yearly or 10-yearly risks of very extreme rainfall events. Three other large areas in the contiguous USA, the Midwest, the Southeast, and Texas, are also studied, and show similar but weaker trends than those in the Northeast.

Nonlinear mixed effects (NLME) modeling based on stochastic differential equations (SDEs) have evolved into a promising approach for analysis of PK/PD data. SDE-NLME models go beyond the realm of standard population modeling as they consider stochastic dynamics, thereby introducing a probabilistic perspective on the state variables. This article presents a summary of the main contributions to SDE-NLME models found in the literature. The aims of this work were to develop an exact gradient version of the first-order conditional estimation (FOCE) method for SDE-NLME models and to investigate whether it enabled faster estimation and better gradient precision/accuracy compared to the use of gradients approximated by finite differences. A simulation-estimation study was set up whereby finite difference approximations of the gradients of each level were interchanged with the exact gradients. Following previous work, the uncertainty of the state variables was accounted for using the extended Kalman filter (EKF). The exact gradient FOCE method was implemented in Mathematica 11 and evaluated on SDE versions of three common PK/PD models. When finite difference gradients were replaced by exact gradients at both FOCE levels, relative runtimes improved between 6- and 32-fold, depending on model complexity. Additionally, gradient precision/accuracy was significantly better in the exact gradient case. We conclude that parameter estimation using FOCE with exact gradients can successfully be applied to SDE-NLME models.