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2013 | OriginalPaper | Buchkapitel

20. Ideal Metrics and Stability of Characterizations of Probability Distributions

verfasst von : Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Erschienen in: The Methods of Distances in the Theory of Probability and Statistics

Verlag: Springer New York

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No probability distribution is a true representation of the probabilistic law of a given random phenomenon: assumptions such as normality, exponentiality, and the like are seldom if ever satisfied in practice.

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Vorheriges Kapitel Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements

Nächstes Kapitel Positive and Negative Definite Kernels and Their Properties

See, for example, Hampel [1971], Huber [1977], Papantoni-Kazakos [1977], and Roussas [1972].

See, for example, Akaike [1981], Csiszar [1967], Kullback [1959], Ljung [1978], and Wasserstein [1969].

See, for example, Zolotarev [1977a,b, 1983], Kalashnikov and Rachev [1985, 1986a,b, 1988], Hernandez-Lerma and Marcus [1984], and Rachev [1989].

See, for example, Cramer [1946, Sect. 18] and Diaconis and Freedman [1987].

See Rachev and Todorovic [1990] and Rachev and Samorodnitsky [1990].

See the problem of stability in risk theory in Sect. 17.2 of Chaps. 17.

For the general case, see Sect. 20.4.

The density of a beta distribution with parameters α and β is given by

$$\frac{\Gamma (\alpha + \beta )} {\Gamma (\alpha )\Gamma (\beta )}{x}^{\alpha -1}{(1 - x)}^{\beta -1},\ 0 < x < 1.$$

See, for example, Feller [1971].

See Cramer [1946, Sect. 18] for the case p = 2.

See, for example, Abramowitz and Stegun [1970, p. 257].

See Cramer [1946, Sect. 18].

Here, as before, $\boldsymbol \rho (X,Y ) :=\mathop{\sup}\limits_{x}\vert F_{X}(x) - F_{Y }(x)\vert $.

See (i) and (ii) in implications (a) and (b) in Sect. 20.1 of this chapter.

See Dudley [2002, Sect. 11.7].

See Definition 15.3.1(i) in Chaps. 15.

See (15.2.1) and (15.2.2) in Chaps. 15.

See (15.2.18) in Chaps. 15.

See (3.3.13) in Chap. 3.

See Remark 19.4.6 in Chaps. 19.

See (20.2.6) with $\mu _{1} = \boldsymbol \rho $.

See (20.2.7) with $\mu _{2} = \boldsymbol \rho $.

See (19.2.4) in Chaps. 19.

In fact, $\boldsymbol \rho _{{_\ast}}$ plays the role of ideal metric for our problem.

See (20.2.2) and Theorems 20.2.1 and 20.2.2.

Rachev and Rüschendorf [1991] discuss the approximate independence of distributions on spheres and their stability properties.

See Definition 15.3.1 in Chap. 15.

See Lemma 20.3.1.

See Abramowitz and Stegun [1970, p. 257].

The basic properties of ℓ _p-metrics were summarized in Chap. 19; see (19.3.9)–(19.3.18).

See Kuelbs [1973] and Samorodnitski and Taqqu [1994].

For other similar examples, see class L, Feller [1971, Sect. 8, Chap. XVII].

See Shiryayev [1984, p. 337].

See Resnick [1987].

See Rachev and Samorodnitsky [1990].

See Sect. 19.4 of Chap. 19.

See Definition 17.4.1 in Chap. 17.

See Kalashnikov and Rachev [1988, Chap. 4, Sect. 2, Lemma 10].

For background on these concepts, see Resnick [1987, Chap. 5].

See Resnick [1987, pp. 257–258].

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Csiszar I (1967) Information-type measures of difference of probability distributions and indirect observations. Studio Sci Math Hungar 2:299–318MathSciNetMATH

Diaconis P, Freedman D (1987) A dozen de Finetti-style results in search of a theory. Ann Inst H Poincare 23:397–623MathSciNetMATH

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Hernandez-Lerma O, Marcus SI (1984) Identification and approximation of Queueing systems. IEEE Trans Automat Contr AC29:472–474

Huber PJ (1977) Robust statistical procedures. CBMS regional conference series in applied mathematics, 27, Society for Industrial and Applied Mathematics, Philadelphia

Kalashnikov VV, Rachev ST (1985) Characterization problems in queueing and their stability. Adv Appl Prob 17:868–886MathSciNetMATHCrossRef

Kalashnikov VV, Rachev ST (1986a) Characterization of inverse problems in queueing and their stability. J Appl Prob 23:459–473MathSciNetMATHCrossRef

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Titel: Ideal Metrics and Stability of Characterizations of Probability Distributions
verfasst von: Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank J. Fabozzi
Verlag: Springer New York
Buch: The Methods of Distances in the Theory of Probability and Statistics
Print ISBN: 978-1-4614-4868-6

Electronic ISBN: 978-1-4614-4869-3

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-1-4614-4869-3_20

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