Lognormal and Gamma Mixed Negative Binomial Regression : Supplementary Material
نویسندگان
چکیده
Proof: The first and second derivatives of f(z) = − ln(1 − pz) are f ′(z) = p 1−pz and f ′′(z) = p (1−pz)2 = (f ′(z))2, respectively, and f j(0) = 0 for j ≥ 1, therefore d dzm f j(z) ∣∣ z=0 = 0 for j > m and thus (2) is true for j > m. When n = j = 1, we have d n dzn f j(z) ∣∣ z=0 = f ′ k(0) = p, ∑n j′=1 F (n, j ′) j! (j−j′!) [f ′(z)]nf j−j ′ (z) ∣∣∣ z=0 = p, and F (n, j)j!pn = p, therefore, (2) is true for n = m = 1. Assume (2) is true for n = m− 1, then
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