In this thesis an optimal stopping problem related to the classical secretary problem is studied. The theory of optimal stopping represents a special branch of stochastic optimization. Here the socalled full information best choice problem with a known number of offers is generalized by maximizing the probability of selecting an r-candidate, where an offer is called r-candidate if it is not lower than the maximal offer reduced by function r. In the first part discrete time is investigated. For this optimal stopping problem to select an r-candidate an optimal stopping time is indicated, the suboptimal myopic stopping time is displayed and threshold rules are studied including asymptotic behaviour. The basis of this optimal stopping problem is displayed in a general setting where the payoff depends on the prophet´s choiceand on the maximal offer, i.e. the value of the prophet. As a further application the mean of the ratio of the gambler´s choice and prophet´s value is investigated. Then in the second part offers arrive in continuous time. Offers are presented according to random arrival times and the horizon terminating the period of choosing is taken to be fixed and random. Here stress is layed on the geometric and on the exponential distribution, i.e. the Poisson process. In the final part the optimal stopping problem of maximizing the duration of owning a sufficiently good offer is applied to the concept of an r-candidate. A distinction between an overall and a temporary r-candidate is made. The duration of owning an r-candidate is investigated for a finite number of offers with regard to recall. The duration problem with discounted epochs is resolved. Finally the duration of owning an r-candidate is considered regarding the Poisson process where the horizon is fixed and exponentially distributed.