Abstract
In this chapter, we discuss the problem of optimal quality usage as a multidimensional Monge–Kantorovich problem. We begin by stating and interpreting the one-dimensional and the multidimensional problems. We provide conditions for optimality and weak optimality in the multivariate case for particular choices of the cost function. Finally, we derive an upper bound for the minimal total losses for a special choice of the cost function and compare it to the upper bound involving the first difference pseudomoment.
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Notes
- 1.
See version (VI) of the Monge–Kantorovich problem in Sect. 5.2 and, in particular, (5.2.36).
- 2.
- 3.
- 4.
- 5.
See Zolotarev [1986, Sect. 1.5].
- 6.
See Case D in Sect. 4.4.
- 7.
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Optimal Quality Usage. In: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4869-3_14
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DOI: https://doi.org/10.1007/978-1-4614-4869-3_14
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