Asymptotic estimators of the sample size in a record model

The connection between Stirling numbers of the first kind and records is well-known. Applying this relationship, we derive bounds for the maximum likelihood estimator of the sample size based on the number of observed records. The proof proceeds by a remarkable expression of the mode of the unsigned Stirling numbers of the the first kind due to Hammersley. Moreover, this representation of the mode leads to an accurate approximation of the maximum likelihood estimator.