Philatelic Mixtures and Multimodal Densities

Abstract A century ago, postage stamps were printed on a deliberate mixture of different paper types, each having its own thickness characteristics due to poor quality control in paper manufacture. It was not unusual for different types of paper to be used for a single stamp issue, because ordinary white wove paper was often not readily available in quantities necessary for production of a new stamp issue. Unfortunately, the different paper types used were not always well documented. Each paper type was purchased by the ream, which was required to consist of a specific number of sheets and had to weigh a certain amount. To cut costs, manufacturers often satisfied both requirements by mixing a few “thick” sheets of paper into an otherwise underweight ream. Although the importance of paper thickness in the philatelic literature is reflected by a higher market value for stamps printed on the thicker and more scarce paper, stamp catalogs have been notoriously vague about characterizing paper thickness, relying instead on classifying stamps by whether they were printed on “thin,” “regular,” or “thick” paper. In this article, the term philatelic mixture is introduced to describe any situation in which a particular stamp issue is known to have been printed on a mixture of paper types with possible differences in paper thickness. In statistical terms, this is a mixture-of-distributions problem, in which identification of the mixture components (different paper types) is achieved using measurements on stamp thickness, and where information regarding the number of such components is vague, at best. The focus of this article is on parametric and nonparametric perspectives of the mixture problem and its special application to identifying the components of philatelic mixtures. The parametric approach tries to fit a finite mixture of component densities of a given type, usually normal, where the number of densities in the mixture may be known or unknown. The nonparametric approach, on the other hand, assumes no such model, but tries to identify components by the resulting placement of modes in the density estimate. A specific example of a philatelic mixture is discussed, the 1872 Hidalgo issue of Mexico, which possesses the aforementioned features, has a large literature of its own, and is a stamp issue for which paper-thickness data were made available to the authors. Nonparametric statistical methods used to analyze these data include nonparametric density estimation and a bootstrap test for multimodality (Silverman 1981a). Various parametric mixture models are also fitted to the stamp-thickness data using a likelihood ratio test (Wolfe 1970), and the results are compared with the nonparametric approach. One of the novel features of the application in this article is the use of extensive historical data to reach conclusions regarding the mixture components.