Lossless Data Embedding For All Image Formats by Fridrich Goljan Du

By Fridrich Goljan Du

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2 Value at Risk and Expected Shortfall For better or worse, Value at Risk (VaR for short) is nowadays a crucial component of most risk analysis/management systems in the financial and insurance industries. Whether this computation is imposed by regulators, or it is done on a voluntary basis by portfolio managers, is irrelevant here. We shall merely attempt to understand the rationale behind this measure of risk. 26 1 UNIVARIATE EXPLORATORY DATA ANALYSIS We introduce the concept at an intuitive level, with the discussion of a simple example of a dynamical model.

We implemented it in S-Plus under the name pot. 2 The Example of the PCS Index It is possible to propose mathematical models for the time evolution of the PCS index. We describe one of them in the Notes & Complements at the end of the chapter. These models are most often quite sophisticated, and they are difficult to fit and use in practice. Instead of aiming at a theory of the dynamics of the index, a less ambitious program is to consider the value of the index on any given day, and to perform a static analysis of its distribution.

Xn are said to be independent if 14 1 UNIVARIATE EXPLORATORY DATA ANALYSIS Fig. 7. Graphical comparison of the Cauchy distribution C(0, 1) and the Gaussian distribution N (0, 1). P{X1 ≤ α1 , X2 ≤ α2 , . . , Xn ≤ αn } = P{X1 ≤ α1 }P{X2 ≤ α2 }P{Xn ≤ αn } for all possible choices of the real numbers α1 , α2 , . . , αn . In other words, the random variables are independent if the joint cdf is the product of the marginal cdf’s. But since such a definition involves multivariate notions introduced in the next chapter, we shall refrain from emphasizing it at this stage, and we shall rely on the intuitive notion of independence.

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