User:Ulf Rehmann/sandbox statprob/

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Table 1. A class of stationary correlation models for longitudinal count data and basic properties.

$$\begin{array}{ccc} \text{Model} & \text{Dynamic relationship} & \text{Mean-variance} \\ && \text{\& Correlations} \\ \hline AR(1) & y_{it}=\rho * y_{i,t-1}+d_{it}, t=2,\ldots & E[Y_{it}]=\mu_{i\cdot} \\ & y_{i1}\sim Poi(\mu_{i\cdot}) & \mbox{var}[Y_{it}]=\mu_{i\cdot} \\ & d_{it} \sim P(\mu_{i\cdot}(1-\rho )), t=2,\ldots & \mbox{corr}[Y_{it},Y_{i,t+\ell}]=\rho_{\ell} \\ && =\rho^{\ell} \\ \hline MA(1) & y_{it}=\rho * d_{i,t-1}+d_{it}, t=2,\ldots & E[Y_{it}]=\mu_{i\cdot} \\ & y_{i1}=d_{i1} \sim Poi(\mu_{i\cdot}/(1+\rho)) & \mbox{var}[Y_{it}]=\mu_{i\cdot} \\ & d_{it} \sim P(\mu_{i\cdot}/(1+\rho )), t=2,\ldots & \mbox{corr}[Y_{it},Y_{i,t+\ell}]=\rho_{\ell} \\ && = \left\{ \begin{array}{ll} \frac{\rho}{1+\rho} & \mbox{for } \ell=1\\ 0 & \mbox{otherwise}, \end{array} \right. \\ \hline EQC & y_{it}=\rho * y_{i1}+d_{it}, t=2,\ldots & E[Y_{it}]=\mu_{i\cdot} \\ & y_{i1}\sim Poi(\mu_{i\cdot}) & \mbox{var}[Y_{it}]=\mu_{i\cdot} \\ & d_{it} \sim P(\mu_{i\cdot}(1-\rho )), t=2,\ldots & \mbox{corr}[Y_{it},Y_{i,t+\ell}]=\rho_{\ell} \\ && =\rho \\ \hline \end{array}$$