APPENDIX DFIRST AND SECOND PARTIAL DERIVATIVES OF THE MIXTURE LOG-LIKELIHOOD FUNCTION

To present the formulas for the first and second partial derivatives of the mixture log-likelihood function (Eq. 4.24) with respect to the five parameters μ, θ, β, γ, and Λ, we initially obtain, from the general form of the likelihood function images

(D.1)images

Then, using the notation fμ(x) = f (x)/μ,  fθ(x) = f (x)/θ, …,

(D.2)images
(D.3)images

and similarly for the partial derivatives of the log-likelihood function with respect to β, γ, and Λ. For the second partial derivatives, using the notation fμμ(x) = 2f (x)/μμ,  fμθ(x) = 2f (x)/μθ, …,

(D.4)images
(D.5)images

and similarly for the other 13 second partial derivatives.

Thus, for ease of presentation and implementation, f(x), S(x), and their first and second partial derivatives with respect to the five parameters can be seen as the building blocks of the first and second partial ...

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