\hfill \thepage} %} \input{tcilatex} \begin{document} \title{A discrete valued time series; The polio dataset} \author{} \maketitle \paragraph{Model description} \bigskip \citeasnoun{zege:1988} analyzed a time series of monthly numbers of poliomyelitis cases during the period 1970--1983 in the US. We make comparison to the performance of the Monte Carlo Newton-Raphson method as reported in \citeasnoun{kuk:chen:1999}. We adopt their model formulation. Let $y_{i}$ denote the number of polio cases in the $i$th period ($% i=1,\ldots ,168)$. It is assumed that the distribution of $y_{i}$ is governed by a latent stationary AR(1) process $\{u_{i}\}$ satisfying \[ u_{i}=\rho u_{i-1}+\varepsilon _{i}, \]% where the $\varepsilon _{i}\sim N(0,\sigma ^{2})$ variables. To account for trend and seasonality the following covariate vector is introduced \[ \mathbf{x}_{i}=\left( 1,\frac{i}{1000},\cos \left( \frac{2\pi }{12}i\right) ,\sin \left( \frac{2\pi }{12}i\right) ,\cos \left( \frac{2\pi }{6}i\right) ,\sin \left( \frac{2\pi }{6}i\right) \right) . \]% Conditionally on the latent process $\{u_{i}\}$, the counts $y_{i}$ are independently Poisson distributed with intensity \[ \lambda _{i}=\exp (\mathbf{x}_{i}^{^{\prime }}\mathbf{\beta }+u_{i}). \] \paragraph{Results} Parameter estimates are shown in the following table. \begin{center} \begin{tabular}{lllllllll} & $\beta _{1}$ & $\beta _{2}$ & $\beta _{3}$ & $\sigma $ & $\kappa _{1}$ & $% \kappa _{1}$ & $\kappa _{1}$ & $\kappa _{1}$ & $\kappa _{1}$ & $\kappa _{1}$ \\ \hline ADMB-RE & 1.9538 & 0.684 & 2.775 & 0.8017 & -4.127 & -2.3909 & 0.4029 & 3.8095 & 2.2254 & 3.2654 \\ aML & 2.064 & 0.688 & 2.841 & 2.2829 & -4.056 & -2.300 & 0.510 & 1.449 & 2.341 & 3.384% \end{tabular} \end{center} We note that not the standard deviation is large for several regression parameters. The ADMB-RE estimates (which are based on the Laplace approximation) very are very similar to the exact maximum likelihood estimates as obtained with the method of \citeasnoun{kuk:chen:1999}. \bibliographystyle{agsm} \bibliography{skaug} \end{document}