By W. J. Thron
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Additional info for Analytic Theory of Continued Fractions II
Here d is the degree of freedom of the multivariate tdistribution. More generally, Heyde (1994, 1997) showed that the REML equations can be derived from a quasi-likelihood. As it turns out, the likelihood under multivariate-t is a special case of Heyde’s quasi-likelihood. For such a reason, the (Gaussian) REML estimation may be regarded as a method of quasi-likelihood. Similarly, the (Gaussian) ML estimation may be justiﬁed from a quasi-likelihood point of view. For simplicity, the corresponding estimators are still called REML or ML estimators.
Vm ), where Vi = Var(yi ). If the Vi s are completely unknown, there are still more unknown covariance parameters than the sample size. In such a case, again, consistent estimation of V is not possible. Therefore, one needs to further specify the Vi s. For simplicity, we assume that for any q, r ∈ Ji , cov(yiq , yir ) does not depend on i. This includes some important cases in practice. The following is an example. 8. Suppose that the observation times are equally spaced. In such a case we may assume, without loss of generality, that tj = j.
A linear mixed model can be expressed as yijk = li + a1 xijk,1 + a2 xijk,2 + sij + eijk , i = 1, . . , 5, j = 1, . . , ni , and k = 1, . . , nij . 1); that is, y = Xβ + Zs + e, where y = (y111 , y121 , . . , y585 ) is the vector of all the observations, β = (l1 , . . , l5 , a1 , a2 ) is the vector of all the ﬁxed eﬀects, X is the matrix of covariates corresponding to β, s = (s11 , s12 , . . , s58 ) is the vector of sire eﬀects, Z is the design matrix corresponding to s, and e = (e111 , e121 , .
Analytic Theory of Continued Fractions II by W. J. Thron