Fisher information in different types of perfect and imperfect ranked set samples from finite mixture models

We derive some general results on the Fisher information (FI) contained in the data obtained from the ranked set sampling (RSS) design relative to its counterpart under the simple random sampling (SRS) for a finite mixture model. We propose different variations of RSS data and show how to calculate the FI matrix for each variation under both perfect and imperfect ranking assumptions. Also, a comparison is made among the proposed variations of RSS data using the missing information criterion. We discuss some interesting cases where the ratio of the determinant of the FI matrices for the RSS and SRS data is independent of the component densities and the number of components of the model and it is always equal to the set size used through the RSS procedure. Theoretical results are augmented by numerical studies for a mixture of two exponential distributions.