By B.D. Craven
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Extra info for Control and Optimization (Applied Mathematics and Mathematical Computation Series)
For the general construction of the PNMLE, numerical algorithms are required. 3 The PNMLE via the EM algorithm A major tool for constructing the maximum likelihood estimates is the EM algorithm (Dempster et al. (1977), McLachlan and Krishnan (1997)). 9) where yij = 1, if center i belongs to subpopulation j, and 0 otherwise. 10) . 11) which can be maximized in θj and qj , separately. This established the M-step of the EM algorithm. In fact, we find easily that qˆj = 1 k k eij . 14). 4 The EMGFU for the profile likelihood mixture When the gradient function indicates heterogeneity, usually the number of components adequate to model this heterogeneity will be unknown and several values for m need to be considered.
Again, convergent † This follows from a second order Taylor expansion of Γ(θ) around θ: ˆ Γ(θ (n) ) ≈ Γ(θ) ˆ + (n) (n) 2 (n) 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ Γ (θ)(θ − θ) + Γ (θ)(θ − θ) /2 = Γ(θ) + Γ (θ)(θ − θ) /2. We assume that ˆ 2 × C which serves as boundedness of the second derivative. 5 behavior appears to be quadratic (a general proof is still outstanding), and it can be expected that the iteration will only need a few steps. Indeed, for this case iteration based upon Γ stops only after 5 steps, whereas the iteration based upon Φ stops after 60 steps.
34) is given on the log-relative risk scale. One ˆ could use the δ-method, so that var(θˆM H ) ≈ (eθM H )2 var(log(θˆM H )), though we prefer to give a more direct comparison. 33) i=1 where αi = ˆ T θn i ˆ T niC +θn i . The Mantel-Haenszel estimate of the common relative T C x n /n T risk is given as θˆM H = i xCi niT /nii with ni = nC i + ni (Greenland and Robins i i i (1985), see also Woodward (1999)). 34) T where xi = xC i + xi as before. 34) has been developed for the situation of identical person-times in the centers reflecting a binomial sampling plan.
Control and Optimization (Applied Mathematics and Mathematical Computation Series) by B.D. Craven