Zero-inflated Conway-Maxwell Poisson Distribution to Analyze Discrete Data

Appendix

(A) Score and Likelihood Ratio Tests

Let Z1,Z2,…,Zn be n rv’s with pmf pz;θ, parameters θ=θ1,θ2,…,θq and Lθ;z be the likelihood function.

The hypothesis of interest

against the alternative

can be tested by using the score and likelihood ratio (LR) tests, where θˆ∗ and θˆ are the maximum likelihood estimates under H0 and H1 respectively. The test statistics of these tests are summarized below.

Likelihood Ratio Test

Score Test

where U∗T=u1θ,u2θ,…,uqθθ=θˆ∗, uiθ=∂lnLθ;z∂θi,i=1,2,…,q

and Γ∗=Γij∗=E−∂2lnLθ;z∂θi∂θjθ=θˆ∗,i,j=1,2,…,q

These test statistics are each asymptotically χ2 distributed with k degrees of freedom.

(B) Derivatives of the CMP pmf and normalizing constant, Zλ,ν

L e t Z i = Z ( λ i , ν )

∂ ln P ( k i  \gt  0 ) ∂ ν = − ln ( k i ! ) − 1 Z i ∂ Z i ∂ ν , ∂ ln P ( k i  \gt  0 ) ∂ β r = k i X i r − 1 Z i ∂ Z i ∂ β r ,

∂ 2 ln P ( k i \gt 0 ) ∂ ν ∂ β r = ∂ Z i ∂ ν ∂ Z i ∂ β r Z i 2 − ∂ 2 Z i ∂ ν ∂ β r Z i , ∂ 2 ln P ( k i  \gt  0 ) ∂ β r ∂ β s = ∂ Z i ∂ β s ∂ Z i ∂ β r Z i 2 − ∂ 2 Z i ∂ β s ∂ β r Z i ,

∂ 2 ln P ( k i > 0 ) ∂ ν 2 = ∂ Z i ∂ ν 2 Z i 2 − ∂ 2 Z i ∂ ν 2 Z i

∂ Z i ∂ β r = ∑ j = 1 ∞ j X i r λ i j ( j ! ) ν , ∂ Z i ∂ ν = ∑ j = 1 ∞ − λ i j ln ( j ! ) ( j ! ) ν ,

∂ 2 Z i ∂ β r ∂ β s = ∑ j = 1 ∞ j 2 X i r X i s λ i j ( j ! ) ν , ∂ 2 Z i ∂ ν ∂ β r = ∑ j = 2 ∞ − j X i r λ i j ln ( j ! ) ( j ! ) ν ,

∂ 2 Z i ∂ ν 2 = ∑ j = 2 ∞ λ i j ln ( j ! ) 2 ( j ! ) ν