Summary: The Generalized Pareto Distribution (GPD) is used for modeling exceedances over thresholds. The general form of the GPD depends on three parameters: the location parameter \(\mu\); the scale parameter (\(\beta >0\)); and the shape parameter (\(-\infty <\xi < \infty\)). This work restricts attention to the case where \(\mu =0\) and shows that, as \(\xi\) decreases while \(\beta\) is kept fixed, the family of \(\mathrm{GPD}(\xi,\beta)\) distributions increases in the usual stochastic order. This property is used for testing the significance of trends in the size of the exceedances over high thresholds in a time series consisting of ozone measurements.