On the nth stop-loss transform order of ruin probability

The concept of the nth stop-loss order is generalized to the class of general nonnegative monotone decreasing functions. The nth stop-loss transform of the first deficit below its initial level is expressed in terms of the nth stop-loss transform of the claim amount random variable. We study the stop-loss ordering of ruin probability through the maximal aggregate loss and obtain a result relating the stop-loss ordering of ruin probabilities to the stop-loss ordering of severities.