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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 diagonal while 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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