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Renshaw and Verrall [ 11 ] specified the generalized linear model (GLM) underlying the chain-ladder technique and suggested some other GLMs which might be useful in claims reserving. The purpose of this paper is to construct bounds for the discounted loss reserve within the framework of GLMs. Exact calculation of the distribution of the total reserve is not feasible, and hence the determination of lower and upper bounds with a simpler structure is a possible way out. The paper ends with numerical examples illustrating the usefulness of the presented approximations. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that the necessary conditions for stability of Kharitonov polynomials can be checked by simply inspecting few upper and lower bound coefficients. [ABSTRACT FROM AUTHOR]

In this paper, the study of best simultaneous Lp-approximation by algebraic polynomials on a union of intervals, I, is considered. We obtain convergence of the best simultaneous approximations when the measure of I tends to zero. We also get an interpolation result. [ABSTRACT FROM AUTHOR]

In this paper, we propose a three-term conjugate gradient method which can produce sufficient descent condition, that is, [image omitted] . This property is independent of any line search used. When an exact line search is used, this method reduces to the standard Hestenes-Stiefel conjugate gradient method. We also introduce two variants of the proposed method which still preserve the sufficient descent property, and prove that these two methods converge globally with standard Wolfe line search even if the minimization function is nonconvex. We also report some numerical experiment to show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of Hankel norm model reduction for linear systems with time-varying delay in the state. For a given stable system, our attention is focused on the construction of reduced-order model, which guarantees the corresponding error system to be asymptotically stable and has a specified Hankel norm error performance. Two different approaches are proposed to solve this problem. One casts the model reduction into a convex optimization problem by using a linearization procedure, and the other is based on the cone complementarity linearization idea, which casts the model reduction into a sequential minimization problem subject to linear matrix inequality constraints. Both continuous and discrete time eases are considered. A numerical example is provided to show the effectiveness of the proposed theory. [ABSTRACT FROM AUTHOR]

Stable indirect and direct adaptive controllers are presented for a class of input-output feedback linearizable timevarying non-linear systems. The radial basis function neural networks are used as on-line approximators to learn the time-varying characteristics of system parameters. Stability results are given in the paper, and the performance of the indirect and direct adaptive schemes is demonstrated through a fault-tolerant engine control problem where the faults are naturally time-varying. [ABSTRACT FROM AUTHOR]

One of the most popular algorithms for solving minimax approximation problems is the Barrodale and Phillips simplex-based method [I. Barrodale and C. Phillips (1974). An improved algorithm for discrete Chebyshev linear approximation. In: Proceedings of the Fourth Manitoba Conference on Numerical Mathematics , pp. 177-190, University of Manitoba, Winnipeg, Canada.]. Our new research has found that the modified pivoting strategy in the first stage of the Barrodale and Phillips algorithm could result in an invalid starting point and system failure in later stages [D. Lei (2002). Robust and efficient algorithms for ℓ 1 and ℓ ∞ approximations. PhD Thesis, School of Computing and Engineering, University of Huddersfield, Huddersfield, UK.]. In contrast, a primal method that we propose in this paper is numerically more stable, and the approximating function does not suffer the effects of wild oscillation. However, on average, the primal method requires more searching steps to find an optimal solution. To improve the efficiency of the primal method, we have considered using the least squares solution to provide a better starting point. Experimental results show that if the least squares estimator is available, then it can speed up the convergence rate significantly in most cases. The number of steps required by using the least squares solution for the primal method also compares favourably with that of the dual method. [ABSTRACT FROM AUTHOR]