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This paper is concerned with the use of the stereographic projection to map the points of a design on the sphere in three dimensions onto a two-dimensional stereogram. Details of the projection and its attendant stereogram are given and the application of the constructs to designs on the sphere is highlighted. Spherical $t$-designs are then introduced and stereograms of selected examples are presented. In addition, examples of regression models with design space the unit ball are considered, stereograms which represent the spherical component of such designs are given and an example in which stereograms are used to elucidate the geometric isomorphism of a suite of designs generated by an exchange algorithm is presented. Emphasis throughout the text is placed on the use of the stereographic projection in teaching and research within the framework of statistics.

This paper is concerned with the formulation of $N$-mixture models for estimating the abundance and probability of detection of a species from binary response, count and time-to-detection data. A modelling framework, which encompasses time-to-first-detection within the context of detection/non-detection and time-to-each-detection and time-to-first-detection within the context of count data, is introduced. Two observation processes which depend on whether or not double counting is assumed to occur are also considered. The main focus of the paper is on the derivation of explicit forms for the likelihoods associated with each of the proposed models. Closed-form expressions for the likelihoods associated with time-to-detection data are new and are developed from the theory of order statistics. A key finding of the study is that, based on the assumption of no double counting, the likelihoods associated with times-to-detection together with count data are the product of the likelihood for the counts alone and a term which depends on the detection probability parameter. This result demonstrates that, in this case, recording times-to-detection could well improve precision in estimation over recording counts alone. In contrast, for the double counting protocol with exponential arrival times, no information was found to be gained by recording times-to-detection in addition to the count data. An R package and an accompanying vignette are also introduced in order to complement the algebraic results and to demonstrate the use of the models in practice., Comment: 21 pages, 1 figure