By Arjun K. Gupta, Tamas Varga, Taras Bodnar
Elliptically Contoured versions in facts and Portfolio conception totally revises the 1st targeted advent to the idea of matrix variate elliptically contoured distributions. There are extra chapters, and all of the unique chapters of this vintage textual content were up to date. assets during this publication might be beneficial for researchers, practitioners, and graduate scholars in information and similar fields of finance and engineering. these attracted to multivariate statistical research and its software to portfolio concept will locate this article instantly worthwhile. In multivariate statistical research, elliptical distributions have lately supplied an alternative choice to the traditional version. Elliptical distributions have additionally elevated their acceptance in finance as a result of the skill to version heavy tails frequently saw in actual info. lots of the paintings, even if, is unfolded in journals during the international and isn't simply obtainable to the investigators. A noteworthy functionality of this publication is the gathering of an important effects at the concept of matrix variate elliptically contoured distributions that have been formerly merely to be had within the journal-based literature. The content material is prepared in a unified demeanour which may serve an a necessary creation to the subject.
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Additional resources for Elliptically Contoured Models in Statistics and Portfolio Theory
PROOF: See Feller (1957), p. 413. 1. Let f be a bounded, not identically zero function defined on IR+ 0 . If f satisfies the equation f (x + y) = f (x) f (y) for all then f (x) = e−kx where k ≥ 0. g. see Anderson and Fang (1982b), Fang and Chen (1984), and Sutradhar and Ali (1989). We will use the following definition given in Gupta and Varga (1994b). 1. Let X be a random matrix of dimensions p × n. ) distribution if its characteristic function has the form φX (T) = etr(iT M)ψ (tr(T Σ TΦ )) with T : p × n, M : p × n, Σ : p × p, Φ : n × n, Σ ≥ 0, Φ ≥ 0, and ψ : [0, ∞) → IR.
T1 , t2 , . . , tn )⎟ ⎝ ⎝ .. ⎠ ⎠ ⎝ 2 ⎝ .. ⎠ ⎠ tn tn n = exp i ∑ t j μ exp − j=1 = n 1 2 n ∑ t jΣ t j ¬ j=1 1 it j μ − t j Σ t j , 2 ∏ exp j=1 which shows that x1 , x2 , . . , xn are independent, each with distribution Np (μ , Σ ). 2 Probability Density Function If X ∼ E p,n (M, Σ ⊗ Φ , ψ ) defines an absolutely continuous elliptically contoured distribution, Σ and Φ must be positive definite. Assume this is not the case. For example, Σ ≥ 0 but Σ is not positive definite. 7, it follows that Σ = GDG where G ∈ O(n), and D is diagonal and d11 = 0.
Then, L0 ,ψ 00 y ∼ E p−q 0, Partition y as y = . 29) y1 , where y1 is k × 1. We have y2 P(x2 − m2 ∈ S) = P(x2 − m2 = Σ 22 a with a ∈ IR p−q ) = P(G(x2 − m2 ) = GΣ 22 G Ga with a ∈ IR p−q ) = P y= L0 b with b ∈ IR p−q 00 = P y= L0 00 b1 b2 with b1 ∈ IRk , b2 ∈ IR p−q−k = P y= Lb1 0 with b1 ∈ IRk y1 y2 =P = c 0 with c ∈ IRk = P(y2 = 0) . 29) that y2 ∼ E p−q−k (0, 0, ψ ) and so P(y2 = 0) = 1. Therefore, P(x2 − m2 ∈ S) = 1. c. distributions. 21. Let X ∼ E p,n (M, Σ ⊗ Φ , ψ ) with stochastic representation M + rAUB .
Elliptically Contoured Models in Statistics and Portfolio Theory by Arjun K. Gupta, Tamas Varga, Taras Bodnar