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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.
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.
Two very commonly used loss functions are the squared loss, () =, and the absolute loss, () = | |.The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case).
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 machine learning and mathematical optimization, loss functions for classification are computationally feasible loss functions representing the price paid for inaccuracy of predictions in classification problems (problems of identifying which category a particular observation belongs to). [1]
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 ...