Abstract
In this chapter, motivated by the seminal paper of Brockett, “Information theoretic approach to actuarial science: A unification and extension of relevant theory and applications,”Transactions of the Society of Actuaries, Vol. 43, 73–135 (1991), we review minimization of the Kullback – Leibler divergence D KL(u,v) between observed (raw) death probabilities or mortality rates, u, and the same entities, v, to be graduated (or smoothed) subject to a set of reasonable constraints such as monotonicity, bounded smoothness, etc. Noting that the quantities u and v, involved in the above minimization problem based on the Kullback – Leibler divergence, are nonprobability vectors, we study the properties of divergence and statistical information theory for D KL(p,q), where pand q are nonprobability vectors. We do the same for the Cressie and Read power divergence between nonprobability vectors, solve the problem of graduation of mortality rates via Lagrangian duality theory, discuss the ramifications of constraints, tests of goodness-of-fit, and compare with other graduation methods, predominantly the Whittaker and Henderson method. At the end we provide numerical illustrations and comparisons.
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Sachlas, A.P., Papaioannou, T. (2010). On a Minimization Problem Involving Divergences and Its Applications. In: Skiadas, C. (eds) Advances in Data Analysis. Statistics for Industry and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4799-5_8
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DOI: https://doi.org/10.1007/978-0-8176-4799-5_8
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