We propose a method of estimating component distribution functions (cdfs) in finite mixture distributions without specifying a parametric form on the true underlying cdfs. As a result, we develop estimators of the component parameters based on these estimated cdfs. This method requires a vector of observations on each subject and involves discretizing the original data into multinomial bins. This results in a mixture of multinomial distributions which has the same mixing proportions as the original mixture. The methods are illustrated on a data set from cognitive psychology.