P Deb, PK. Trivedi/Journal of Health Economics 21(2002)601-625 2.3. LCM In the lCm, the random variable is postulated as a draw from a population which is ar dditive mixture of C distinct subpopulations in proportions T1,..., Tc, where 2i=Ij 1,j200=1,., C). The mixture density for observation i,i=I,., n, is given by f(:1)=∑xf(v0)+xcf(1c),i=1,…,n (28) where each term in the sum on the right-hand side is the product of the mixing probability T and the component(subpopulation) density f; (ilej). The j are unknown constants that are estimated along with all other parameters, denoted 0. Also Ic=(1-2=lI).For identification( normalization), we use the labelling restriction that T≥m2≥…≥rc which can always be satisfied by rearrangement post estimation The component densities of the C-component finite mixture are specified as 入 f (iles) 厂(吵)rO+1)(入,+yj where j=1, 2, .., C are the latent classes, Aj. i exp(x' Bi)and vj. i=(1/a j)k.Note that(Bj, aj )are unrestricted across components The conditional mean of the count variable is given by E(ylx)=A=∑可 (2.10) and the variance by Vyx)=∑x+1+一对 (211) Both the mean and the variance in the lcm are in gener erent from their standard NB counterparts. The LCM can also accommodate over and underdispersed data relative to the NBM, but does so in a different manner than the TPM. Because this density is very flexible(permitting, for example, multimodal marginal distributions) and easily captures long-right tails, it is likely to accommodate patterns of overdispersion expected in our data 2. 4. Properties oflcM On LCMs offer a flexible way of specifying mixtures of densities. There are a number advantages of using a discrete rather than a continuous mixing distribution. First, the finite mixture representation provides a natural and intuitively attractive representation of 2 In general, T; may be parameterized as a function of covariates. However, such models are often fraught with identification problems if separating information is not available. However, if separating information is available, identification is feasible(Duan et al., 1983)606 P. Deb, P.K. Trivedi / Journal of Health Economics 21 (2002) 601–625 2.3. LCM In the LCM, the random variable is postulated as a draw from a population which is an additive mixture of C distinct subpopulations in proportions π1,... , πC, where C j=1 πj = 1, πj ≥ 0 (j = 1,... , C). The mixture density for observation i, i = 1,... , n, is given by f (yi|θ) = C −1 j=1 πjfj (yi|θj ) + πCfC(yi|θC), i = 1, . . . , n, (2.8) where each term in the sum on the right-hand side is the product of the mixing probability πj and the component (subpopulation) density fj (yi|θj ). The πj are unknown constants that are estimated along with all other parameters, denoted θ. 2 Also πC = (1−C−1 j=1 πj ). For identification (normalization), we use the labelling restriction that π1 ≥ π2 ≥ ··· ≥ πC, which can always be satisfied by rearrangement post estimation. The component densities of the C-component finite mixture are specified as fj (yi|θj ) = Γ (yi + ψj,i) Γ (ψj,i)Γ (yi + 1)
ψj,i λj,i + ψj,i ψj,i
λj,i λj,i + ψj,i yi , (2.9) where j = 1, 2,... , C are the latent classes, λj,i = exp(x iβj )and ψj,i = (1/αj )λk j,i. Note that (βj , αj )are unrestricted across components. The conditional mean of the count variable is given by E(yi|xi) = λ¯i =  C j=1 πjλji (2.10) and the variance by V(yi|xi) =  C j=1 πjλ2 ji[1 + αjλ−k ji ] + λ¯i − λ¯ 2 i . (2.11) Both the mean and the variance in the LCM are, in general, different from their standard NB counterparts. The LCM can also accommodate over and underdispersed data relative to the NBM, but does so in a different manner than the TPM. Because this density is very flexible (permitting, for example, multimodal marginal distributions) and easily captures long-right tails, it is likely to accommodate patterns of overdispersion expected in our data. 2.4. Properties of LCM LCMs offer a flexible way of specifying mixtures of densities. There are a number of advantages of using a discrete rather than a continuous mixing distribution. First, the finite mixture representation provides a natural and intuitively attractive representation of 2 In general, πj may be parameterized as a function of covariates. However, such models are often fraught with identification problems if separating information is not available. However, if separating information is available, identification is feasible (Duan et al., 1983).