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The chain-ladder or development [1] method is a prominent [2] [3] actuarial loss reserving technique. The chain-ladder method is used in both the property and casualty [1] [4] and health insurance [5] fields. Its intent is to estimate incurred but not reported claims and project ultimate loss amounts. [5]
It is primarily used in the property and casualty [5] [9] and health insurance [2] fields. Generally considered a blend of the chain-ladder and expected claims loss reserving methods, [2] [8] [10] the Bornhuetter–Ferguson method uses both reported or paid losses as well as an a priori expected loss ratio to arrive at an ultimate loss estimate.
Ultimate loss amounts are necessary for determining an insurance company's carried reserves. They are also useful for determining adequate insurance premiums, when loss experience is used as a rating factor [4] [5] [6] Loss development factors are used in all triangular methods of loss reserving, [7] such as the chain-ladder method.
In population genetics, a triangle plot of genotype frequencies is called a de Finetti diagram. In game theory [2] and convex optimization, [3] it is often called a simplex plot. In a ternary plot, the values of the three variables a, b, and c must sum to some constant, K. Usually, this constant is represented as 1.0 or 100%.
In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. [4]
In Python, the function cholesky from the numpy.linalg module performs Cholesky decomposition. In Matlab, the chol function gives the Cholesky decomposition. Note that chol uses the upper triangular factor of the input matrix by default, i.e. it computes = where is upper triangular. A flag can be passed to use the lower triangular factor instead.
The union of the tree and the matching is a cycle, with no possible shortcuts, and with weight approximately 3n/2. However, the optimal solution uses the edges of weight 1 + ε together with two weight-1 edges incident to the endpoints of the path, and has total weight (1 + ε)(n − 2) + 2, close to n for small values of ε. Hence we obtain an ...
Note that the intermediate points that were constructed are in fact the control points for two new Bézier curves, both exactly coincident with the old one. This algorithm not only evaluates the curve at , but splits the curve into two pieces at , and provides the equations of the two sub-curves in Bézier form.