New improved estimators for overdispersion in models with clustered multinomial data and unequal cluster sizes

Abstract

It is usual to rely on the quasi-likelihood methods for deriving statistical methods applied to clustered multinomial data with no underlying distribution. Even though extensive literature can be encountered for these kind of data sets, there are few investigations to deal with unequal cluster sizes. This paper aims to contribute to fill this gap by proposing new estimators for the intracluster correlation coefficient.

Appendix

1.1 Zero-inflated binomial distribution

The binomial distribution with zero inflation in the first cell, i.e., n-inflation in the second cell, is given by

$$\begin{aligned}&\left( \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=v\right) \\&\quad =\left\{ \begin{array}{lll} \mathcal {M}\left( n, \begin{pmatrix} p_{1}(\varvec{\theta })\\ p_{2}(\varvec{\theta }) \end{pmatrix} \right) , &{} \quad \text {if }v=1, &{} \text {with }\Pr (V=1)=w\\ n\varvec{e}_{2}, &{} \quad \text {if }v=0, &{} \text {with }\Pr (V=0)=1-w \end{array} \right. . \end{aligned}$$

Its first order moment vector is given by

$$\begin{aligned} E\left[ \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right]&=E\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] \\&=E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=1\right] \Pr \left( V=1\right) \\&\quad +E\left[ \left. n\varvec{e}_{2}\right| V=0\right] \Pr \left( V=0\right) \\&=n \begin{pmatrix} wp_{1}(\varvec{\theta })\\ 1-wp_{1}(\varvec{\theta }) \end{pmatrix} . \end{aligned}$$

The derivation for the the second order moment matrix calculation is given by

$$\begin{aligned} E\left[ Var\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right]&=Var\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=1\right] \Pr \left( V=1\right) \\&\quad +Var\left[ \left. n\varvec{e}_{2}\right| V=0\right] \Pr \left( V=0\right) \\&=Var\left[ \mathcal {M}\left( n, \begin{pmatrix} p_{1}(\varvec{\theta })\\ p_{2}(\varvec{\theta }) \end{pmatrix} \right) \right] w\\&=nwp_{1}(\varvec{\theta })\left( 1-p_{1}(\varvec{\theta })\right) \begin{pmatrix} 1 &{}\,\, -1\\ -1 &{}\,\, 1 \end{pmatrix}, \end{aligned}$$

$$\begin{aligned}&Var\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] \\&=E\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] E^{T}\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] \\&\quad -E\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] E^{T}\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] \\&=E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=1\right] E^{T}\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=1\right] w\\&\quad +E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=0\right] E^{T}\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V=0\right] (1-w)\\&\quad -n^{2} \begin{pmatrix} wp_{1}(\varvec{\theta })\\ 1-wp_{1}(\varvec{\theta }) \end{pmatrix} \begin{pmatrix} wp_{1}(\varvec{\theta })&1-wp_{1}(\varvec{\theta }) \end{pmatrix} \\&=n^{2}(1-w)wp_{1}^{2}(\varvec{\theta }) \begin{pmatrix} 1 &{}\quad -1\\ -1 &{}\quad 1 \end{pmatrix} , \end{aligned}$$

and hence

$$\begin{aligned}&Var\left[ \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right] \\&\quad =E\left[ Var\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] +Var\left[ E\left[ \left. \begin{pmatrix} Y_{1}\\ Y_{2} \end{pmatrix} \right| V\right] \right] \\&\quad =nwp_{1}(\varvec{\theta })\left[ \left( 1-p_{1}(\varvec{\theta })\right) +n(1-w)p_{1}(\varvec{\theta })\right] \begin{pmatrix} 1 &{}\quad -1\\ -1 &{}\quad 1 \end{pmatrix} \\&\quad =nwp_{1}(\varvec{\theta })(1-wp_{1}(\varvec{\theta }))(1+\rho ^{2}(n-1)) \begin{pmatrix} 1 &{}\quad -1\\ -1 &{}\quad 1 \end{pmatrix} , \end{aligned}$$

where

$$\begin{aligned} \rho ^{2}=\frac{(1-w)p_{1}(\varvec{\theta })}{1-wp_{1}(\varvec{\theta } )},\quad \text {for any }w\in (0,1). \end{aligned}$$

This result matches the one given in Morel and Neerchal (2012, p. 83). Let

$$\begin{aligned}&\left( \left. \varvec{Y}\right| V=v\right) \nonumber \\&\quad =\left\{ \begin{array}{l@{\quad }l@{\quad }l} \mathcal {M}\left( n,\varvec{p}(\varvec{\theta })\right) , &{} \text {if }v=1, &{} \text {with }\Pr (V=1)=w\\ n\varvec{e}_{M}, &{} \text {if }v=0, &{} \text {with }\Pr (V=0)=1-w \end{array} \right. \end{aligned}$$

be the multinomial distribution with zero inflation in the first $M-1$ cells, i.e., n-inflation in the M-th cell.

For $M\ge 3$, a univariate homogeneous intracluster correlation coefficient, $\rho ^{2}$, seems not to be an appropriate measure to characterize the variability of this distribution, since the intracluster correlation along the cells seems to be heterogeneous. The reason for this is that for $M\ge 3$ there is not an expression for the variance-covariance matrix of the multinomial distribution defined as a matrix not depending on parameters multiplied by a scalar with all the information about the parameters of the distribution.

1.2 Proof of Theorem 3.2

Let

$$\begin{aligned} \varvec{S}_{\varvec{Y}}=\frac{1}{N-1}\sum _{\ell =1}^{N}\left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p}}\right) \left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p}}\right) ^{T}, \end{aligned}$$

the matrix of quasi-variances and quasi-covariances of the simple random sample $\varvec{Y}^{(1)},\ldots ,\varvec{Y}^{(N)}$ and

$$\begin{aligned} \overline{\varvec{S}}_{\varvec{Y}}&=\mathrm {diag}(\varvec{S} _{\varvec{Y}})= \begin{pmatrix} S_{Y_{1}}^{2} &{} &{} \\ &{} \ddots &{} \\ &{} &{} S_{Y_{M}}^{2} \end{pmatrix} ,\\ S_{Y_{r}}^{2}&=\frac{1}{N-1}\sum _{\ell =1}^{N}(Y^{(\ell ,r)}-n\widehat{p} _{r})^{2}. \end{aligned}$$

It is well-known that each diagonal element of $\overline{\varvec{S} }_{\varvec{Y}}$ is a consistent estimator of each diagonal element of $\vartheta _{n}n\varvec{\Sigma }_{\varvec{p}(\varvec{\theta })}$, i.e.,

$$\begin{aligned} \mathrm {E}\left[ \overline{\varvec{S}}_{\varvec{Y}}\right] =\mathrm {diag}\{\mathrm {E}\left[ \varvec{S}_{\varvec{Y}}\right] \}= & {} \mathrm {diag}\{\mathrm {Var}[\varvec{Y}^{(\ell )}]\}\\= & {} \mathrm {diag} \{\vartheta _{n}n\varvec{\Sigma }_{\varvec{p}(\varvec{\theta })}\}, \end{aligned}$$

and

$$\begin{aligned}&S_{Y_{r}}^{2}\overset{P}{\underset{N\rightarrow \infty }{\longrightarrow } }\vartheta _{n}np_{r}(\varvec{\theta })\left( 1-p_{r}(\varvec{\theta })\right) ,\quad r=1,\ldots ,M,\nonumber \\&\text {or}\quad \overline{\varvec{S}}_{\varvec{Y}} \overset{P}{\underset{N\rightarrow \infty }{\longrightarrow }}\mathrm {diag} (\vartheta _{n}n\varvec{\Sigma }_{\varvec{p}(\varvec{\theta } )}). \end{aligned}$$

(8.1)

It is not difficult to establish that

$$\begin{aligned} \mathrm {trace}(\overline{\varvec{S}}_{\varvec{Y}})= & {} \sum _{r=1} ^{M}S_{Y_{r}}^{2}=\mathrm {trace}(\varvec{S}_{\varvec{Y}})\nonumber \\= & {} \frac{1}{N-1}\sum _{\ell =1}^{N}\left( \varvec{Y}^{(\ell )} -n\widehat{\varvec{p}}\right) ^{T}\left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p}}\right) , \end{aligned}$$

(8.2)

which is consistent for $\mathrm {trace}(\vartheta _{n}n\varvec{\Sigma }_{\varvec{p}(\varvec{\theta })})=\vartheta _{n}n\sum _{r=1}^{M} p_{r}(\varvec{\theta })\left( 1-p_{r}(\varvec{\theta })\right)$. We know that the chi-square test-statistic $X^{2}(\widetilde{\varvec{Y}})$, given in (3.3), has an asymptotic $\mathcal {\chi }_{(N-1)(M-1)}^{2}$ distribution for fixed values of number of clusters N and an increasing cluster size, n, under the assumption of inter-cluster level homogeneity. However, this distribution is not a useful device for the proof. Based on the expression of the chi-square test-statistic, $X^{2}(\widetilde{\varvec{Y} })$, in terms of the variance-covariance matrix, as well as the same steps to obtain the expression and consistency of (8.2), we are going to establish (3.4). We have

$$\begin{aligned}&\mathrm {trace}(\overline{\varvec{S}}_{\varvec{Y}}\tfrac{1}{n}\varvec{D}_{\varvec{p}(\varvec{\theta })}^{-1})\\&\quad =\frac{1}{N-1}\sum _{\ell =1}^{N}\left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p} }\right) ^{T}\tfrac{1}{n}\varvec{D}_{\varvec{p}(\varvec{\theta } )}^{-1}\left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p}}\right) \end{aligned}$$

and

$$\begin{aligned}&\mathrm {E}\left[ \mathrm {trace}(\overline{\varvec{S}}_{\varvec{Y} }\tfrac{1}{n}\varvec{D}_{\varvec{p}(\varvec{\theta })} ^{-1})\right] \\&\quad =\mathrm {traceE}\left[ \overline{\varvec{S} }_{\varvec{Y}}\tfrac{1}{n}\varvec{D}_{\varvec{p} (\varvec{\theta })}^{-1}\right] =\mathrm {trace}\left( \mathrm {E}\left[ \overline{\varvec{S}}_{\varvec{Y}}\right] \tfrac{1}{n}\varvec{D} _{\varvec{p}(\varvec{\theta })}^{-1}\right) \\&\quad =\mathrm {trace}\left( \vartheta _{n}n\varvec{\Sigma }_{\varvec{p}(\varvec{\theta })} \tfrac{1}{n}\varvec{D}_{\varvec{p}(\varvec{\theta })}^{-1}\right) \\&\quad =\vartheta _{n}\mathrm {trace}\left( \varvec{\Sigma }_{\varvec{p} (\varvec{\theta })}\varvec{D}_{\varvec{p}(\varvec{\theta } )}^{-1}\right) \\&\quad =\vartheta _{n}\mathrm {trace}\left( \left( \varvec{D} _{\varvec{p}(\varvec{\theta })}-\varvec{p}(\varvec{\theta })\varvec{p}^{T}(\varvec{\theta })\right) \varvec{D} _{\varvec{p}(\varvec{\theta })}^{-1}\right) \\&\quad =\vartheta _{n}\left[ \mathrm {trace}(\varvec{I}_{M})-\mathrm {trace} (\varvec{p}(\varvec{\theta })\varvec{1}_{M}^{T})\right] =\vartheta _{n}(M-1). \end{aligned}$$

Hence,

$$\begin{aligned}&\mathrm {E}\left[ \frac{1}{M-1}\mathrm {trace}(\overline{\varvec{S} }_{\varvec{Y}}\tfrac{1}{n}\varvec{D}_{\varvec{p} (\varvec{\theta })}^{-1})\right] \\&\quad =\mathrm {E}\Bigg [ \frac{1}{(N-1)(M-1)}\sum _{\ell =1}^{N}\left( \varvec{Y}^{(\ell )} -n\widehat{\varvec{p}}\right) ^{T}\\&\qquad \times \tfrac{1}{n}\varvec{D} _{\varvec{p}(\varvec{\theta })}^{-1}\left( \varvec{Y}^{(\ell )}-n\widehat{\varvec{p}}\right) \Bigg ] =\vartheta _{n}, \end{aligned}$$

and taking into account that $\widehat{\varvec{p}}$ is a consistent estimator of $\varvec{p}(\varvec{\theta })$, as $N\rightarrow \infty$, as well as (8.1),

$$\begin{aligned}&\frac{1}{M-1}\mathrm {trace}(\overline{\varvec{S}}_{\varvec{Y}} \tfrac{1}{n}\varvec{D}_{\widehat{\varvec{p}}}^{-1})\\&\quad =\frac{1}{(N-1)(M-1)}\sum _{\ell =1}^{N}\left( \varvec{Y}^{(\ell )} -n\widehat{\varvec{p}}\right) ^{T}\tfrac{1}{n}\varvec{D} _{\widehat{\varvec{p}}}^{-1}\left( \varvec{Y}^{(\ell )} -n\widehat{\varvec{p}}\right) \\&\quad =\frac{X^{2}(\widetilde{\varvec{Y}} )}{(N-1)(M-1)} \end{aligned}$$

tends in probability to $\vartheta _{n}$, as $N\rightarrow \infty$. In other words,

$$\begin{aligned}&\frac{X^{2}(\widetilde{\varvec{Y}})}{(N-1)(M-1)}\\&\quad =\frac{1}{(M-1)n} \sum _{r=1}^{M}\frac{1}{\widehat{p}_{r}}S_{Y_{r}}^{2}\overset{P}{\underset{N\rightarrow \infty }{\longrightarrow }}\frac{\vartheta _{n}n}{(M-1)n}\\&\qquad \sum _{r=1}^{M}\frac{p_{r}(\varvec{\theta })}{p_{r} (\varvec{\theta })}\left( 1-p_{r}(\varvec{\theta })\right) =\vartheta _{n}. \end{aligned}$$

In addition, taking into account (1.9), the right hand size of (3.4) follows. Finally, we like to mention that even though $X^{2}(\widetilde{\varvec{Y}})$ and $\vartheta _{n}(N-1)(M-1)$ have the same expectation for a fixed value of N, this proof is not trivial since $\vartheta _{n}(N-1)(M-1)$ as well as $X^{2}(\widetilde{\varvec{Y}})$ tend to infinite as $N\rightarrow \infty$.

1.3 Proof of Theorem 2.2

By applying the Central Limit Theorem it holds (3.1). Hence, from Pardo (2006, formula (7.10)), for the minimum phi-divergence estimator of $\varvec{\theta }$ of a log-linear model it holds

$$\begin{aligned} \sqrt{N}(\widehat{\varvec{\theta }}_{\phi }-\varvec{\theta }_{0})= & {} \left( \varvec{\varvec{W}}^{T}\varvec{\Sigma \varvec{_{\varvec{p} \left( \theta _{0}\right) }}W}\right) ^{-1}\varvec{W}^{T} \varvec{\Sigma }_{p\left( \varvec{\theta }_{0}\right) }\nonumber \\&\times \,\varvec{D} _{\varvec{p}\left( \theta _{0}\right) }^{-1}\sqrt{N}\left( \widehat{\varvec{p}}-\varvec{p}\left( \varvec{\theta }_{0}\right) \right) +o_{p}\left( \varvec{1}_{M_{0}}\right) , \end{aligned}$$

(8.3)

and the variance-covariance matrix of $\sqrt{N}(\widehat{\varvec{\theta } }_{\phi }-\varvec{\theta }_{0})$ is

$$\begin{aligned}&\tfrac{\vartheta _{n}}{n}\left( \varvec{\varvec{W}}^{T} \varvec{\Sigma \varvec{_{\varvec{p}\left( \theta _{0}\right) }} W}\right) ^{-1}\varvec{W}^{T}\varvec{\Sigma }_{p\left( \varvec{\theta }_{0}\right) }\varvec{D}_{\varvec{p}\left( \theta _{0}\right) }^{-1}\nonumber \\&\qquad \times \varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }\varvec{D}_{\varvec{p}\left( \theta _{0}\right) }^{-1}\varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }\varvec{W}\left( \varvec{\varvec{W}}^{T}\varvec{\Sigma \varvec{_{\varvec{p} \left( \theta _{0}\right) }}W}\right) ^{-1}\nonumber \\&\quad =\tfrac{\vartheta _{n}}{n}\left( \varvec{\varvec{W}}^{T} \varvec{\Sigma \varvec{_{\varvec{p}\left( \theta _{0}\right) }} W}\right) ^{-1}. \end{aligned}$$

(8.4)

The last equality comes from

$$\begin{aligned} \varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }\varvec{D}_{\varvec{p}\left( \theta _{0}\right) }^{-1} \varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }=\varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }. \end{aligned}$$

From the Taylor expansion of $\varvec{p}(\widehat{\varvec{\theta } }_{\phi })$ around $\varvec{p}(\varvec{\theta }_{0})$ we obtain

$$\begin{aligned} \sqrt{N}(\varvec{p}(\widehat{\varvec{\theta }}_{\phi })-\varvec{p} (\varvec{\theta }_{0}))=\varvec{\Sigma \varvec{_{\varvec{p} \left( \theta _{0}\right) }}W}\sqrt{N}(\widehat{\varvec{\theta }}_{\phi }-\varvec{\theta }_{0}) +o_{p}\left( \varvec{1}_{M}\right) , \end{aligned}$$

(8.5)

and the variance-covariance matrix of $\sqrt{N}(\varvec{p} (\widehat{\varvec{\theta }}_{\phi })-\varvec{p}(\varvec{\theta } _{0}))$ is

$$\begin{aligned} \tfrac{\vartheta _{n}}{n}\varvec{\Sigma \varvec{_{\varvec{p}\left( \theta _{0}\right) }}W}\left( \varvec{\varvec{W}}^{T} \varvec{\Sigma \varvec{_{\varvec{p}\left( \theta _{0}\right) }} W}\right) ^{-1}\varvec{W}^{T}\varvec{\Sigma }_{p\left( \varvec{\theta }_{0}\right) }. \end{aligned}$$

(8.6)

Since $\sqrt{N}\left( \widehat{\varvec{p}}-\varvec{p}\left( \varvec{\theta }_{0}\right) \right)$ is normal and centred, from (8.3) and (8.4), (2.8) is obtained. Similarly, since $\sqrt{N}(\widehat{\varvec{\theta }}_{\phi }-\varvec{\theta }_{0})$ is normal and centred, from (8.5) and (8.6), (2.9) is obtained.

1.4 Derivation of Formula (4.4)

Multiplying (4.3) by $\sqrt{N_{g}}n_{g}\big / \sum \limits _{h=1}^{G}n_{h}N_{h}$

$$\begin{aligned} w_{g}(\widehat{\varvec{p}}^{(g)}-\varvec{p}\left( \varvec{\theta }_{0}\right) )\overset{\mathcal {L}}{\underset{N_{g}\rightarrow \infty }{\longrightarrow }}\mathcal {N}\left( \varvec{0}_{M},\tfrac{n_{g} N_{g}\vartheta _{n_{g}}}{\left( \sum \nolimits _{h=1}^{G}n_{h}N_{h}\right) ^{2}}\varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta } _{0}\right) }\right) , \end{aligned}$$

hence summing up from $g=1$ to G and by the independence of clusters

$$\begin{aligned}&\sum \limits _{g=1}^{G}w_{g}(\widehat{\varvec{p}}^{\left( g\right) }-\varvec{p}\left( \varvec{\theta }_{0}\right) )\\&\quad =\left( \widehat{\varvec{p}}-\varvec{p}\left( \varvec{\theta }_{0}\right) \right) \\&\qquad \overset{\mathcal {L}}{\underset{N_{g}\rightarrow \infty ,\;g=1,\ldots ,G}{\longrightarrow }}\mathcal {N}\left( \varvec{0}_{M} ,\tfrac{\sum \nolimits _{g=1}^{G}n_{g}N_{g}\vartheta _{n_{g}}}{\left( \sum \nolimits _{h=1}^{G}n_{h}N_{h}\right) ^{2}}\varvec{\Sigma }_{\varvec{p}\left( \varvec{\theta }_{0}\right) }\right) . \end{aligned}$$

Finally multiplying the previous expression by $\sum \nolimits _{h=1} ^{G}n_{h}N_{h}\big / \sqrt{\sum \nolimits _{g=1}^{G}n_{g}N_{g}\vartheta _{n_{g}}}$, the desired expression is obtained.

1.5 Algorithms for Dirichlet-multinomial, n-inflated and random-clumped distributions

The usual parameters of the M-dimensional random variable $\varvec{Y} =(Y_{1},\ldots ,Y_{M})^{T}$ with Dirichlet-multinomial distribution are $\varvec{\alpha }=\left( \alpha _{11},\ldots ,\alpha _{M1}\right) ^{T}$, where $\alpha _{r1}=\frac{1-\rho ^{2}}{\rho ^{2}}p_{r}\left( \varvec{\theta }\right) , r=1,\ldots ,M$. For convenience it is considered with parameters $\varvec{\beta }= {\begin{pmatrix} \rho ^{2}\\ \varvec{p}(\varvec{\theta }) \end{pmatrix}} , \varvec{p}\left( \varvec{\theta }\right) =\left( p_{1}\left( \varvec{\theta }\right) ,\ldots ,p_{M}\left( \varvec{\theta }\right) \right) ^{T}$, and is generated as follows:

STEP 1. :

Generate $B_{1}\sim Beta(\alpha _{11},\alpha _{12})$, with $\alpha _{11}=\frac{1-\rho ^{2}}{\rho ^{2}}p_{1}\left( \varvec{\theta }\right) , \alpha _{12}=\frac{1-\rho ^{2}}{\rho ^{2}}(1-p_{1}\left( \varvec{\theta }\right) )$.

STEP 2. :

Generate $\left( Y_{1} |B_{1}=b_{1}\right) \sim Bin(n,b_{1})$.

STEP 3. :

For $r=2,\ldots ,M-1$ do:

Generate $B_{r}\sim Beta(\alpha _{r1},\alpha _{r2})$ , with $\alpha _{r1} =\frac{1-\rho ^{2}}{\rho ^{2}}p_{r}\left( \varvec{\theta }\right) , \alpha _{r2}=\frac{1-\rho ^{2}}{\rho ^{2}}\left( 1-\sum _{h=1}^{r} p_{h}\left( \varvec{\theta }\right) \right)$.

Generate $( Y_{r}|Y_{1}=y_{1},\ldots ,Y_{r-1} =y_{r-1},B_{r}=b_{r}) \sim Bin\left( n-\sum _{h=1}^{r-1} y_{h},b_{r}\right)$.

STEP 4. :

Do $\left( Y_{M}|Y_{1}=y_{1},\ldots ,Y_{M-1}=y_{M-1}\right) =n-\sum _{h=1}^{M-1}y_{h}$.

The random variable $\varvec{Y}=(Y_{1},\ldots ,Y_{M})^{T}$ of the n-inflated multinomial distribution with parameters $\varvec{\beta }, \varvec{p}\left( \varvec{\theta }\right)$, is generated as follows:

STEP 1. :

Generate $V\sim Ber(\rho ^{2})$.

STEP 2. :

Generate

$$\begin{aligned}&\left( \varvec{Y|}V=v\right) =\left\{ \begin{array}{l@{\quad }l} \mathcal {M}(n,\varvec{p}\left( \varvec{\theta }\right) ), &{} \text {if }v=0\\ n\mathcal {M}(1,\varvec{p}\left( \varvec{\theta }\right) ), &{} \text {if }v=1 \end{array} \right. . \end{aligned}$$

The random variable $\varvec{Y}=(Y_{1},\ldots ,Y_{M})^{T}$ of the random clumped distribution with parameters $\varvec{\beta }, \varvec{p} \left( \varvec{\theta }\right)$, is generated as follows:

STEP 1. :

Generate $\varvec{Y}_{0}=(Y_{01} ,\ldots ,Y_{0M})^{T}\sim \mathcal {M}(1,\varvec{p}\left( \varvec{\theta }\right) )$.

STEP 2. :

Generate $K_{1}\sim Bin(n,\rho )$.

STEP 3. :

Generate $\left( \varvec{Y}_{1}|K_{1}=k_{1}\right) =\big ( (Y_{11},\ldots ,Y_{1M})^{T} | K_{1}=k_{1}\big ) \sim \mathcal {M}(n-k_{1},\varvec{p}\left( \varvec{\theta }\right) )$.

STEP 4. :

Do $\left( \varvec{Y|}K_{1}=k_{1}\right) \varvec{=Y}_{0}k_{1}+\left( \varvec{Y}_{1}|K_{1}=k_{1}\right)$.

For the details about the equivalence of this algorithm and (1.12), see Morel and Nagaraj (1993).

It is interesting to note that there exists the package “Modeling overdispersion in R” useful to generate the distributions considered in this Appendix. For more details see Raim et al. (2015).