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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Consider a finite set of points, located on the circumference of a circle. Several tests have been proposed of the hypothesis that the points constitute a random sample from a uniform distribution. In this paper we study a test statistic defined as the maximal number of points that can be covered by some semicircle. Exact and asymptotic distributions under the null hypothesis, and under a certain alternative hypothesis, are given together with some tables. A related test statistic is studied briefly. An expression is obtained concerning most powerful invariant tests of the hypothesis of a uniform circular distribution.

In this paper some results are given on the addivity of chain-ladder projections. Given two claims development triangles, when do their chain-ladder projections add up to the projections of the combined triangle, that is the triangle being the element-wise sum of the two given triangles?Necessary and sufficient conditions for equality are given. These are of a fairly simply form and are directly connected to the ordinary chain-ladder calculations. In addition, sufficient conditions of the same form are given for inequality between the combined projection vector and the sum of the two original projections vectors.

The multiplicative ratemaking, model we have in mind is the following one. Within a certain branch of insurance we have, say for simplicity, two tarif arguments U and V. For example, in motor insurance we could think of U and V as being make of car and geographical district respectively. In fire insurance U could be class of construction for buildings and V could relate to fire defense capacities.The arguments are of a qualitative nature and argument U has r levels, while argument V has k levels. To our disposal we have statistical experience of the business for a certain period of time, consisting of—risk exposures nij (i = 1 … r, j = 1 … k).Risk exposure nij thus corresponds to the ith U-level and the jth V-level. It could be e.g. number of policy years or sum insured during the period of observation for objects belonging simultaneously to U-level i and V-level j.The nijS are known non-random quantities.—(relative) risk measures pij(i = 1 … r, j = 1 …k).Risk measure pij could be e.g. claims frequency, i.e. number of Claims divided by number of policy years, or claims cost per policy year or claims cost as a percentage of sum insured. In general pij is thus the observed number or the observed amount of claims belonging simultaneously to U-level i and V-level j, divided by the corresponding risk exposure nij.

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