multivariate normal distribution theorem relationship between normal and noncentral kh distributions let (uin be an -dimensional normal random vector and let vk vm be linear subspaces of dimensions and mrespectivelywith let xk and xm be orthogonal projections of onto vk and vm and let uk and um be the corresponding projections of thenthe following holds the random vectors xk xm xk and xm are independent kxk kh (kuk )kxm xk kh - (kum uk )and kx xm kh - (ku um prooflet vn be an orthonormal basis of rn such that vk spans vk and vm spans vm by ( we can write the orthogonal projection matrices onto pj vi > kmnwhere vn is defined as rn note that pn is simply the idenas = tity matrix let :[ vn and define :[ zn ]vx recall from section that any orthogonal transformation such as vx is length preservingthat iskzk kxk to prove the first statement of the theoremnote that vx vp [ ] km it follows that (xm xk [ zk+ zm ]and ( xm [ zm+ zn ]moreoverbeing linear transformation of normal random vectorz is also normalwith covariance matrix vv in in particularthe {zi are independent this shows that xk xm xk and xm are independent as well nextobserve that kxk kvxk kz kwhere :[ zk ]the latter vector has independent components with variances and its squared norm has therefore (by definitiona kh (thdistribution the noncentrality parameter is th kez kexk kuk kagain by the length-preserving property of orthogonal transformations this shows that kxk kh (kuk kthe distributions of kxm xk and kx xm follow by analogy theorem is frequently used in the statistical analysis of normal linear modelssee section in typical situations lies in the subspace vm or even vk -in which case kxm xk kh - and kx xm kh - independently the (scaledquotient then turns out to have an distribution - consequence of the following theorem theorem relationship between kh and distributions let kh and kh be independent thenu/ (mnv/ prooffor notational simplicitylet / and / the pdf of / is given by fw ( fu (wvv fv (vdv substituting the pdfs of the corresponding gamma