Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-17T16:31:30.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Isotonic Optimization in Tariff Construction

Published online by Cambridge University Press:  29 August 2014

W.S. Jewell
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of the best fit to set ideal values under general inequality order restrictions is examined for asymmetric, quadratic, absolute, and Chebyshev norms. Special solution procedures are given in terms of network flow algorithms over a network associated with the given isotonic order relations, and the nature of the optimal solutions is characterized for the different norms.

The model is formulated in terms of finding an optimal insurance rate structure over given risk classes for which a desired pattern of tariffs can be specified. The suitability of different norms is considered in the context of corporate profitability, and the relationship to a simple rate relativities model is described.

Keywords

isotonic RegressionLinear ProgrammingQuadratic Programming Network Flows Approximation Theory Insurance Ratemaking
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 8 , Issue 2 , September 1975 , pp. 175 - 203
Copyright
Copyright © International Actuarial Association 1975

References

[1]Bailey, R.and Simon, L., “Two Studies in Automobile Insurance Ratemaking,” ASTIN Bulletin, Vol.I, Part IV, pp. 192–217, (December 1960).CrossRefGoogle Scholar
[2]Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D., Statistical Inference under Order Restrictions, Wiley and Sons, New York, 388 pp., (1972).Google Scholar
[3]Barlow, R. E. and Ubhaya, V. A., “Isotonic Approximation,” in Proceedings of the Symposium on Optimizing Methods in Statistics, Academic Press, New York.Google Scholar
[4]Barrowdale, I., “On Computing Best L 1 Approximations,” in Approximation Theory, (Talbot, A., ed.), Academic Press, 356 pp., (1970).Google Scholar
[5]Barrowdale, I.and Young, A., “Algorithms for Best L 1 and L ∞, Linear Approximations as a Discrete Set,” Num. Math., Vol. 8, pp. 295–306, (1966).CrossRefGoogle Scholar
[6]Boot, J. C. G., Quadratic Programming, North-Holland, 210 pp., (1964).Google Scholar
[7]Cryer, J. D., Robertson, T., Wright, F.T. and Casody, R.J., “Monotone Median Regression,” Dept. of Statistics, University of Iowa, (August 1972).CrossRefGoogle Scholar
[8]Dennis, J. B., Mathematical Programming and Electrical Networks, J. Wiley and Sons, 186 pp. (1959).Google Scholar
[9]Dorn, W. S., “Duality in Quadratic Programming,” Quant. Appl. Math., Vol. 18, pp. 155–162, (1960).CrossRefGoogle Scholar
[10]Duffin, R. J. and Kartovitz, L. A., “Formulation of Linear Programs in Analysis in Approximation Theory,” SIAM J. Appl. Math., Vol. 16, No. 4, pp. 662–675, (July 1968).CrossRefGoogle Scholar
[11]Ford, L. R. Jr., and Fulkerson, D. R., Flows in Networks, Princeton University Press, 1974 PP. (1962).Google Scholar
[12]Golub, G. H. and Saunders, M. A., “Linear Least Squares and Quadratic Programming,” Chapter 10 in Integer and Nonlinear Programming, (Abadie, J., ed.), North-Holland, 544 pp., (1970).Google Scholar
[13]Jung, J., “On Automobile Insurance Ratemaking,” ASTIN Bulletin, Vol. V. Part I, pp. 41–48, (May 1968).CrossRefGoogle Scholar
[14]Kelley, J. E. Jr., “An Application of Linear Programming to Curve Fitting,” J. Soc. Indust. Appl. Math., Vol. 6, No. 1, pp. 15–22, (March 1958).CrossRefGoogle Scholar
[15]Künzi, H.P., Tzschach, H. G., and Zehnder, C. A., Numerical Methods of Mathematical Optimization, Academic Press, 219 pp., (1971).Google Scholar
[16]Lewis, J. T., “Approximation with Convex Constraints,” SIAM Review, Vol. 15, No. 1, pp. 193–217, (January 1973).CrossRefGoogle Scholar
[17]Rabenowitz, P., “Mathematical Programming and Approximation,” in Approximation Theory (Talbot, A., ed.), Academic Press, 356 pp., (1970).Google Scholar
[18]Rabenowitz, P., “Applications of Linear Programming to Numerical Analysis,” SIAM Review, Vol. 10, No. 2, pp. 121–159, (April 1968).CrossRefGoogle Scholar
[19]Rice, J. R., The Approximation of Functions, Vol. 1: Linear Theory, Addison-Wesley, 203 pp., (1964).Google Scholar
[20]Schmitter, H. and Straub, E., “Quadratic Programming in Insurance,” ASTIN Bulletin, Vol.8, No. 3, pp. 311–322, (March 1974).CrossRefGoogle Scholar
[21]Simmonard, M., Linear Programming, Prentice-Hall, 430 pp., (1966).Google Scholar
[22]Stiefel, E., “Note on Jordan Elimination, Linear Programming and Tchebycheff Approximation,” Num. Math., Vol. 2, pp. 1–17, (1960).CrossRefGoogle Scholar
[23]Ubhaya, V.A., “Isotonic and Convex Optimization,” ORC Report 7113, Operations Research Center, University of California, Berkeley, (July 1971).Google Scholar
[24]Veinott, A.F. Jr., “Least d-Majorized Network Flows with Inventory and Statistical Applications,” Management Science, Vol.17, No. 9, pp. 547–567. (May 1971).CrossRefGoogle Scholar
[25]Wagner, H. M., “Linear Programming Techniques for Regression Analysis,” Journal Amer. Stat. Assoc., Vol.54, pp. 206–212, (1959).CrossRefGoogle Scholar
[26]Wagner, H. M., “Non-Linear Regression with Minimal Assumptions,” Journal Amer. Stat. Assoc., Vol. 57, No. 299, pp. 573–578, (September 1962).CrossRefGoogle Scholar