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Last week, we've introduced the concept of exchangeable variables, i.e. satisfying for any matrix , i.e. is a permutation matrix: belongs to the orthogonal group, , and with elements in . It is possible to extend that family, considering all matrices in the orthogonal group, i.e. for all . Since orthogonal matrices can be seen as **Freakonometrics - Tag - R-english**, and kindly contributed to R-bloggers)*rotation*matrices, it will mean, e.g. that density is invariant by rotations,So level curves will be circles (in dimension 2), or more generally spheres. This will yield the concept of spherical distribution (or

*spherically contoured distributions*), that will be extended to elliptical distributions (see e.g. Hartman & Wintner (1940), Kelker (1970) or Cambanis, Huang & Simons (1979))

- spherically contoured distributions

*generator*, and we say that . Equivalently, has a spherical distribution if . A popular example is the Gaussian distribution (centered, with independent margins)Note that there exist a nice stochastic representation of spherically contoured distribution, where is a positive random variable, independent of , uniformly distributed over the unit sphere of , i.e. This construction can be related to the following decomposition

- from circles to ellipses

*only*the variance of the first component (above), while if we change the variance of the second one (below)If we change

*only*the correlation, the axis of the ellipse are still the first and the second diagonalwhile the impact of correlation when X and Y do not have the same variance gives us the following transformations,

- elliptically contoured distributions

> library(mnormt) > x <- seq(-2,4,length=21) > mu <- c(1,3,2) > Sigma <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) > df <- 4 > x=c(0,0);y=c(0,1); z=c(0,2) > dmt(cbind(x,y,z), mu, Sigma,df) [1] 0.006957689 0.020602030 > rmt(n=5, mu, Sigma, df) [,1] [,2] [,3] [1,] 0.42210352 2.7539135 1.659392 [2,] 1.07968146 -0.1364883 4.851956 [3,] -0.04107115 1.6163407 4.123731 [4,] 0.19784451 2.9329165 1.013374 [5,] 1.13456027 0.4737548 -2.054909

To

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