Recursions for distribution functions and stop-loss transforms.

For any functions on the non-negative integers, we can evaluate the cumulative function given by (s) = sx=o(x) from the values of by the recursion (s) = (s - 1) + (s). Analogously we can use this procedure t times to evaluate the t-th order cumulative function t when itself satisfies a certain sort of recursion. We shall also derive recursions for the tth order tails t where (s) = x=s+1(x). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer (1981). We discuss in particular convolutions and compound functions.Distribution; Functions;

Some distributional properties of a class of counting distributions with claims analysis applications

We discuss a class of counting distributions motivated by a problem in discrete surplus analysis, and special cases of which have applications in stop-loss, discrete Tail value at risk (TVaR) and claim count modelling. Explicit formulas are developed, and the mixed Poisson case is considered in some detail. Simplifications occur for some underlying negative binomial and related models, where in some cases compound geometric distributions arise naturally. Applications to claim count and aggregate claims models are then given.published_or_final_versio

On some properties of a class of multivariate Erlang mixtures with insurance applications

We discuss some properties of a class of multivariate mixed Erlang distributions with different scale parameters and describes various distributional properties related to applications in insurance risk theory. Some representations involving scale mixtures, generalized Esscher transformations, higher-order equilibrium distributions, and residual lifetime distributions are derived. These results allows for the study of stop-loss moments, premium calculation, and the risk allocation problem. Finally, some results concerning minimum and maximum variables are derived and applied to pricing joint life and last survivor policies.postprin

A note on deficit analysis in dependency models involving Coxian claim amounts

postprin

Effects of fluoxetine on functional outcomes after acute stroke (FOCUS): a pragmatic, double-blind, randomised, controlled trial

Background Results of small trials indicate that fluoxetine might improve functional outcomes after stroke. The FOCUS trial aimed to provide a precise estimate of these effects. Methods FOCUS was a pragmatic, multicentre, parallel group, double-blind, randomised, placebo-controlled trial done at 103 hospitals in the UK. Patients were eligible if they were aged 18 years or older, had a clinical stroke diagnosis, were enrolled and randomly assigned between 2 days and 15 days after onset, and had focal neurological deficits. Patients were randomly allocated fluoxetine 20 mg or matching placebo orally once daily for 6 months via a web-based system by use of a minimisation algorithm. The primary outcome was functional status, measured with the modified Rankin Scale (mRS), at 6 months. Patients, carers, health-care staff, and the trial team were masked to treatment allocation. Functional status was assessed at 6 months and 12 months after randomisation. Patients were analysed according to their treatment allocation. This trial is registered with the ISRCTN registry, number ISRCTN83290762. Findings Between Sept 10, 2012, and March 31, 2017, 3127 patients were recruited. 1564 patients were allocated fluoxetine and 1563 allocated placebo. mRS data at 6 months were available for 1553 (99·3%) patients in each treatment group. The distribution across mRS categories at 6 months was similar in the fluoxetine and placebo groups (common odds ratio adjusted for minimisation variables 0·951 [95% CI 0·839–1·079]; p=0·439). Patients allocated fluoxetine were less likely than those allocated placebo to develop new depression by 6 months (210 [13·43%] patients vs 269 [17·21%]; difference 3·78% [95% CI 1·26–6·30]; p=0·0033), but they had more bone fractures (45 [2·88%] vs 23 [1·47%]; difference 1·41% [95% CI 0·38–2·43]; p=0·0070). There were no significant differences in any other event at 6 or 12 months. Interpretation Fluoxetine 20 mg given daily for 6 months after acute stroke does not seem to improve functional outcomes. Although the treatment reduced the occurrence of depression, it increased the frequency of bone fractures. These results do not support the routine use of fluoxetine either for the prevention of post-stroke depression or to promote recovery of function. Funding UK Stroke Association and NIHR Health Technology Assessment Programme

Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts

Gerber-Shiu analysis with the generalized penalty function proposed by Cheung et al. (in press-a) is considered in the Sparre Andersen risk model with a Kn family distribution for the interclaim time. A defective renewal equation and its solution for the present Gerber-Shiu function are derived, and their forms are natural for analysis which jointly involves the time of ruin and the surplus immediately prior to ruin. The results are then used to find explicit expressions for various defective joint and marginal densities, including those involving the claim causing ruin and the last interclaim time before ruin. The case with mixed Erlang claim amounts is considered in some detail. © 2009 Elsevier B.V. All rights reserved.link_to_subscribed_fulltex

On the class of erlang mixtures with risk theoretic applications

A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.link_to_subscribed_fulltex

On the analysis of a general class of dependent risk processes

A generalized Sparre Andersen risk process is examined, whereby the joint distribution of the interclaim time and the ensuing claim amount is assumed to have a particular mathematical structure. This structure is present in various dependency models which have previously been proposed and analyzed. It is then shown that this structure in turn often implies particular functional forms for joint discounted densities of ruin related variables including some or all of the deficit at ruin, the surplus immediately prior to ruin, and the surplus after the second last claim. Then, employing a fairly general interclaim time structure which involves a combination of Erlang type densities, a complete identification of a generalized Gerber-Shiu function is provided. An application is given applying these results to a situation involving a mixed Erlang type of claim amount assumption. Various examples and special cases of the model are then considered, including one involving a bivariate Erlang mixture model. © 2012 Elsevier B.V.link_to_subscribed_fulltex

Recursions for distribution functions and stop-loss transforms

For any functions on the non-negative integers, we can evaluate the cumulative function given by (s) = sx=o(x) from the values of by the recursion (s) = (s - 1) + (s). Analogously we can use this procedure t times to evaluate the t-th order cumulative function t when itself satisfies a certain sort of recursion. We shall also derive recursions for the tth order tails t where (s) = x=s+1(x). The recursions can be applied for exact and approximate evaluation of distribution functions and stop-loss transforms of probability distributions. The class of recursions for includes the classes discussed by Sundt (1992), incorporating the class studied by Panjer (1981). We discuss in particular convolutions and compound functions.status: publishe