Some results on the compound Markov binomial model |
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Authors: | Kam-Chuen Yuen Junyi Guo |
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Affiliation: | 1. Department of Statistics and Actuarial Science , The University of Hong Kong , Pokfulam Road, Hong Kong E-mail: kcyuen@hku.hk kcyuen@hku.hk;3. School of Mathematical Sciences and LPMC , Nankai University , Tianjin, 300071, P. R, China E-mail: jyguo@nankai.edu.cn |
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Abstract: | Abstract Some years ago, in the course of an analysis of upper and lower limits for incomplete moments of statistical distributions I established an elementary summation formula1 which proved rather useful for the purpose I had in view. Subsequently the formula was generalized by professor Steffensen, who showed2 that the formula in question could be looked upon as giving the first term of an expansion in a certain type of series. Professor Steffensen established recurrence formulae for the coefficients of the series and computed the second, third and fourth term and the corresponding remainders1, but did not arrive at a general, explicite expression for the coefficient of the n-th term and the corresponding remainder. A year later I found these expressions accidentally while I was working on some other problem. I also discovered the real nature of the procedure in question which proved to be a certain kind of least square fitted polynomial approximation. I did not, however, at the time publish the result. Taking the question up again later I found that the whole problem could be considerably generalized. The type of generalization in question is analogous to the generalization from polynomials to arbitrary functions. |
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Keywords: | Compound Markov binomial semi-Markov expected discounted penalty function ruin probability deficit at ruin surplus just prior to ruin |
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