Expansions of E(XZ|Y+ϵX) and their applications to the analysis of elliptically contoured measures
✍ Scribed by P.J. Szabłowski
- Publisher
- Elsevier Science
- Year
- 1990
- Tongue
- English
- Weight
- 437 KB
- Volume
- 19
- Category
- Article
- ISSN
- 0898-1221
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✦ Synopsis
Almraet--We present a general theorem concerning the relationship between the first two conditional moments of some random variables conditioned upon a random vector and the distribution of the conditioning random vector.
We apply this result to study properties of marginal densities of elliptieally contoured measures and their relation to conditional variances.
Note on the notation used
All vectors considered are columns, x r denotes transposition of vector x, i.e. x r is a row vector. Matrices and vectors are multiplied in a usual way, hence in particular xrAx, where x ~ R n is vector while A is n x n matrix, is a quadratic form.
x i denotes usually ith coordinate of the vector x.
If X e R n is random vector then EX E R n is a vector of expectations of coordinates of X. cov(X, Y)= E(X-EX)(Y-EY) r for random vectors X, Y cov(X, X)~ V(X).
Similar denotation is used in the ease of conditional expectation and covariance. I(A) denotes index function of the set A.