New PDF release: Elliptically Contoured Models in Statistics and Portfolio

By Arjun K. Gupta, Tamas Varga, Taras Bodnar

ISBN-10: 1461481538

ISBN-13: 9781461481539

Elliptically Contoured versions in facts and Portfolio thought absolutely revises the 1st unique creation to the speculation 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 can be worthwhile for researchers, practitioners, and graduate scholars in statistics and comparable fields of finance and engineering. these attracted to multivariate statistical research and its software to portfolio concept will locate this article instantly worthy. ​In multivariate statistical research, elliptical distributions have lately supplied an alternative choice to the traditional version. Elliptical distributions have additionally elevated their attractiveness in finance as a result of the skill to version heavy tails often saw in genuine information. lots of the paintings, although, is opened up in journals through the international and isn't simply obtainable to the investigators. A noteworthy functionality of this ebook is the gathering of an important effects at the conception of matrix variate elliptically contoured distributions that have been formerly basically to be had within the journal-based literature. The content material is equipped in a unified demeanour that may serve an a priceless creation to the topic.

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Additional info for Elliptically Contoured Models in Statistics and Portfolio Theory

Example text

The characteristic function of y is φy (t) = IR pn exp(it y)h(y y)dy , where t ∈ IR pn . Next, we prove that if t1 and t2 are vectors of dimension pn such that t1 t1 = t2 t2 , then φy (t1 ) = φy (t2 ). 11, we see that there exists H ∈ O(pn), such that t1 H = t2 . Therefore, φy (t2 ) = = IR pn IR pn exp(it2 y)h(y y)dy exp(it1 Hy)h(y y)dy . Let z = Hy. The Jacobian of the transformation y → z is |H | pn = 1. So IR pn exp(it1 Hy)h(y y)dy = = IR pn IR pn exp(it1 z)h(z HH z)dz exp(it1 z)h(z z)dz = φy (t1 ).

Let X ∼ E p,n (0, I p ⊗ In , ψ ) with stochastic representation X ≈ rU. 5 Stochastic Representation 33 ⎛ ⎞ X1 ⎜ X2 ⎟ ⎜ ⎟ X=⎜ . ⎟, ⎝ .. ⎠ Xm where Xi is pi × n matrix, i = 1, . . , m. Then, ⎞ ⎛ ⎞ rr1 U1 X1 ⎜ X2 ⎟ ⎜ rr2 U2 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ . ⎟≈⎜ . ⎟, ⎝ .. ⎠ ⎝ .. ⎠ ⎛ rrm Um Xm where r, (r1 , r2 , . . , rm ), U1 , U2 , . . , Um are independent, ri ≥ 0, i = 1, . . , m, 2 ∑m i=1 ri = 1, 2 (r12 , r22 , . . 11) and vec(Ui ) is uniformly distributed on S pi n , i = 1, 2, . . , m. PROOF: Since X ≈ rU, we have ⎛ ⎞ X1 ⎜ X2 ⎟ ⎜ ⎟ ⎜ .

Now, x2 ∼ E p−q (m2 , Σ 22 , ψ ) and so x2 − m2 ∼ E p−q (0, Σ 22 , ψ ). L0 Let k = rk(Σ 22 ). Let G ∈ O(p − q) such that GΣ 22 G = , where L is a 00 diagonal and nonsingular k × k matrix and define y = G(x2 − m2 ). 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) .

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Elliptically Contoured Models in Statistics and Portfolio Theory by Arjun K. Gupta, Tamas Varga, Taras Bodnar


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