Elliptically Contoured Distributions

Transcription

1 Elliptically Contoured Distributions Recall: if X N p µ, Σ), then { 1 f X x) = exp 1 } det πσ x µ) Σ 1 x µ) So f X x) depends on x only through x µ) Σ 1 x µ), and is therefore constant on the ellipsoidal surfaces x µ) Σ 1 x µ) = k. Each ellipsoid is centered at µ and its principal axes are in the directions of the eigenvectors of Σ, with lengths proportional to eigenvalues of Σ. NC STATE UNIVERSITY 1 / 8

2 The general elliptically contoured distribution has pdf 1 g [ x ν) Λ 1 x ν) ] det Λ where: Λ is positive definite; g ) 0, with... g y y) dy 1 dy... dy p = 1. If X has this distribution and Y = L 1 X ν), where LL = Λ, then Y has pdf g y y), spherically contoured. NC STATE UNIVERSITY / 8

3 Polar Coordinates Every y can be written in polar coordinates as y 1 = r sin θ 1, y = r cos θ 1 sin θ, y 3 = r cos θ 1 cos θ sin θ 3,. y p 1 = r cos θ 1 cos θ... cos θ p sin θ p 1, y p = r cos θ 1 cos θ... cos θ p cos θ p 1 where: r 0; 1 π < θ i 1 π, 1 i p 1, and π < θ p 1 π. NC STATE UNIVERSITY 3 / 8

4 The Jacobian Jr, θ) = r p 1 [cos θ 1 )] p [cos θ )] p 3... [cos θ p 3 )] cos θ p ) If the pdf of Y is g y y), then the pdf of R, Θ is Jr, θ)g r ) = r p 1 g r ) [cos θ 1 )] p [cos θ )] p 3... [cos θ p 3 )] cos θ p ) R and Θ are independent, and the marginal pdf of R is π 1 p Γ 1 p) r p 1 g r ) = Cp) r p 1 g r ). NC STATE UNIVERSITY 4 / 8

5 Here Cp) is the surface area of the p-dimensional unit sphere. The vector U = Y/R is uniformly distributed on that surface, and, because it is a function of Θ, is independent of R. So Y can be written as RU, where R and U are independent, R has density Cp) r p 1 g r ), and U is uniformly distributed on the surface of the p-dimensional unit sphere. NC STATE UNIVERSITY 5 / 8

6 Moments If O is an orthogonal matrix O O = I p ), and V = OU then V V = U U = 1, so V also takes values on the surface of the unit sphere. Also the Jacobian is det O = 1, so V is uniformly distributed on the surface of the p-dimensional unit sphere, the same as U. So EU) = EOU) = OEU) for every orthogonal O EU) = 0 take O = I p, for instance). NC STATE UNIVERSITY 6 / 8

7 Similarly, if E UU ) = Σ, then Σ = E VV ) = OΣO for every orthogonal O Σ = ki p for some k not so trivial... )... and U U = 1 kp = traceσ) = E [ trace UU )] = E U U ) = 1 Σ = E UU ) = p 1 I p. NC STATE UNIVERSITY 7 / 8

8 Finally: if ER) <, then EY) = ERU) = ER)EU) = 0; if E R ) <, then E YY ) = E R UU ) = E R ) E UU ) = 1 p E R ) I p and since X = LY + ν, under the same conditions EX) = ν and CX) = LCY)L = 1 p E R ) LL = 1 p E R ) Λ NC STATE UNIVERSITY 8 / 8

9 Marginal Distributions If Y is spherically contoured with density g y y), and Y is partitioned as ) Y 1) Y = Y ) where Y 1) contains Y 1, Y,..., Y q, and Y ) contains Y q+1, Y q+,..., Y p, the marginal density of Y ) is g q+1):p y q+1, y q+,..., y p ) =... g u u + y y ) du = Cq) 0 = g y y ), say. g r 1 + y y ) r q 1 1 dr 1 NC STATE UNIVERSITY 9 / 8

10 So the marginal distribution of Y ) is spherically contoured. Now suppose that X has the elliptically contoured density 1 g [ x ν) Λ 1 x ν) ] det Λ and is similarly partitioned, and as for the multivariate normal distribution we let Z 1) = X 1) Σ 1 Σ 1 X) = X 1) Λ 1 Λ 1 X) Z ) = X ) NC STATE UNIVERSITY 10 / 8

11 Then the density of Z is [ 1 g z 1) τ 1)) Λ 1 det 11 z 1) τ 1)) Λ11 det Λ + z ) ν )) Λ 1 z ) ν ))] where τ 1) = ν 1) Λ 1 Λ 1 ν) Note that Z 1) and Z ) are uncorrelated, but independent only for the multivariate normal distribution. NC STATE UNIVERSITY 11 / 8

12 The marginal density of Z ) = X ) is [ 1 g x ) ν )) Λ 1 x ) ν ))] det Λ = Cq) 0 g [ r1 + x ) ν )) Λ 1 x ) ν ))] r q 1 1 dr 1 So the marginal density of X ) is also elliptically contoured. NC STATE UNIVERSITY 1 / 8

13 Uniform Distribution on a Sphere Consider the special case Y = pu, uniformly distributed on the sphere of radius p. Then Y 1 = pu 1 = p sin Θ 1 ), and the pdf of Θ 1 is proportional to [cos θ 1 )] p 1. So the pdf of Y 1 and hence of every Y i ) is proportional to 1 y p ) p 3 NC STATE UNIVERSITY 13 / 8

14 This is the Beta density with parameters α = β = p 1)/, scaled to the interval [ p, p]. Special cases: p = 1: the density isn t integrable, so the argument fails. Clearly Y 1 takes the values ±1 with probability 1. p = : the density is U-shaped, unbounded at ±. p = 3: the density is uniform on ± 3. p : the density converges to N0, 1). NC STATE UNIVERSITY 14 / 8

15 More generally, for any fixed q, the marginal density of Y 1) consisting of Y 1, Y,..., Y q ) converges to N q 0, I q ) as p. If the radius is a random R p instead of the fixed p, the density of Y 1) is the corresponding mixture of scaled distributions, and as p converges to the corresponding mixture of normal distributions. So if Y is spherically contoured because it is part of a larger random vector that is also spherically contoured, of arbitrarily large dimension, its distribution must be a mixture of spherical normal distributions. NC STATE UNIVERSITY 15 / 8

16 Conditional Distributions The conditional density of Y 1) given Y ) = y is g y 1 y 1 + y y ) g y y = g y 1 y 1 + r ) ) ) g r where r = y y. This is a spherically contoured density, so Y 1) has conditional mean zero and conditional covariance matrix C Y 1) Y ) = y )) = 1 q E Y 1) Y 1) ) R = r I q NC STATE UNIVERSITY 16 / 8

17 After simplification, E Y 1) Y 1) ) R = r = 0 0 r q+1 1 g r 1 + r ) dr1 r q 1 1 g r 1 + r ) dr1 In general, this depends on the conditional value r. It does not depend on r for the multivariate normal, because in that case g r1 + r ) g r ) ) = 1 g r g0) NC STATE UNIVERSITY 17 / 8

18 The conditional density of X 1) given X ) = x ) is { 1 [ ) g x 1) ν 1) B x ) ν ))] g r Λ 1 det 11 Λ11 [ x 1) ν 1) B x ) ν ))] } + r where r = B = Λ 1 Λ 1 x ) ν )) Λ 1 x ) ν )), NC STATE UNIVERSITY 18 / 8

19 This conditional density is also elliptically contoured, with E X 1) X ) = x )) = ν 1) + B x ) ν )) and C X 1) X ) = x )) = 1 q E R 1 R = r ) Λ11 The linearity of the conditional mean is shared with the multivariate normal distribution. The conditional covariance matrix is in general not constant, the exception being the multivariate normal distribution. NC STATE UNIVERSITY 19 / 8

20 Example: Multivariate t Suppose that Z N p 0, I p ), and ms χ m, independent of Z. Then Y = 1/s)Z has the multivariate t density with m degrees of freedom, and Γ m+p ) Γ ) m mπ) p/ R p = Y Y p ) 1 + y y m+p m F p,m = χ p/p χ m/m Every normalized linear combination c Y with c c = 1 has the univariate t-distribution with m degrees of freedom. NC STATE UNIVERSITY 0 / 8

21 Note: if m >, E R /p ) = m/m ), so CY) = m m I p. The standardized multivariate t-distribution is defined by Y s = m m m Y = ms so that E Y sy s /p) = 1 and CY s ) = I p. The standardized multivariate t-distribution is convenient for simulating long-tailed data with a given mean vector and covariance matrix, but it is not the familiar version. Z NC STATE UNIVERSITY 1 / 8

22 For the multivariate t-distribution, the conditional density of Y 1) given Y ) = y is also multivariate t, with m = m + p q) degrees of freedom, but with a scale factor that depends on r = y y. Conditionallly on Y ) = y, Y 1) = 1 Y 1) 1 + r p q) m has the conventional multivariate t-distribution in q dimensions with m degrees of freedom. Since this distribution does not depend on y, Y 1) is independent of Y ). NC STATE UNIVERSITY / 8

23 So if m >, C ) Y 1) Y ) = y = = [ 1 + r [ 1 + r = m + r m = m + r m I q. ] p q) C m ] p q) m p q) Y 1) m m I q I q ) Y ) = y That is, large values of Y ) in magnitude) increase the conditional covariance matrix of Y 1), and small values decrease it. NC STATE UNIVERSITY 3 / 8

24 The reason is that the magnitude of Y ) provides information about s : conditionally on Y ) = y, m + r ) s χ m. Heuristically, a large value of Y ) suggests a small value of s, which in turn implies a large value of Y 1). Similarly, a small value of Y ) suggests a large value of s, which in turn implies a small value of Y 1). NC STATE UNIVERSITY 4 / 8

25 For the general elliptically contoured multivariate t-distribution with m degrees of freedom, the density function is Γ m Γ m+p ) ) mπ) p/ det Λ [ 1 + x µ) Λ 1 ] m+p x µ) m The conditional density of X 1) given X ) = x is correspondingly multivariate t-distribution with m degrees of freedom. NC STATE UNIVERSITY 5 / 8

26 The conditional distribution is centered at µ 1) + B x ) µ )) where B = Λ 1 Λ 1. If m > 1 as it must be), this is the conditional mean. NC STATE UNIVERSITY 6 / 8

27 The matrix of the quadratic form in the conditional density is where now m + r m Λ 11 r = x ) µ )) Λ 1 If m >, the conditional covariance matrix is x ) µ )) m + r m Λ 11 NC STATE UNIVERSITY 7 / 8

28 The conditional structure of the multivariate t-distribution differs from that of the multivariate normal distribution in a surprisingly simple way: The conditional mean is the same linear function of x ) ; The conditional covariance matrix is multiplied by a scalar) quadratic function of x ), instead of being constant. NC STATE UNIVERSITY 8 / 8