= Grouped Poisson likelihood
{c}
{title2=$\ell(\theta)=\sum_{i=0}^{r-1}n_i\log p_i(\theta)+n_r\log\mathbb P_\theta(X\geq r)$}
When a <Poisson distribution> is observed in cells $0,1,\ldots,r-1,\geq r$, the last cell probability is a tail sum, not the probability of exactly $r$. With $p_i(\theta)=e^{-\theta}\theta^i/i!$ and $Q_r(\theta)=1-\sum_{i<r}p_i(\theta)$, telescoping derivatives give $Q_r'=p_{r-1}$. Thus an interior <maximum-likelihood estimate> solves $\sum_{i<r}n_i(i/\theta-1)+n_rp_{r-1}(\theta)/Q_r(\theta)=0$. Substitution of the estimate into these cell probabilities supplies fitted expected counts for a <likelihood-ratio test>. With $r+1$ cells and one regular fitted parameter, the asymptotic <chi-squared distribution> has $r-1$ degrees of freedom. Replacing every grouped tail observation by $r$ changes the data model and generally does not give the grouped maximum likelihood estimate.
Back to article page
