Nomination sampling is a sampling process in which every observation is the maximum of a random sample from some population. Assuming that all samples are taken from a single underlying distributionF, data may be viewed as consisting of pairs (Xi,Ki), whereKiis the size of theith sample and, givenKi=ki, Xiis distributed according toFki. Willemain (1980), who discussed nomination sampling in the context of health care delivery, proposed an estimator for the median ofFunder the assumption thatKi=N, a fixed integer. In this article, the assumption of a fixed sample sizeNfrom each population is relaxed; withKtaken as random, the problem of nonparametric estimation of the distribution functionFis considered. The nonparametric maximum likelihood estimator ofFis derived, its consistency is demonstrated, and its asymptotic behavior as a stochastic process is identified. Conditions are given under which these asymptotic results hold for nonrandomK. A by-product of this development is the consistency of Willemain's estimator of the median. Several applications are considered. For example, nomination sampling arises naturally in certain reliability experiments; the applicability of the derived estimator in problems involving designed redundancy is noted. A detailed analysis of data on 34 yearly maximum floods of the Nidd River is presented, and the estimator of the underlying flood distributionFis displayed.