Codex @codex  0

This is not a valid ordinary R glm family specification. A multinomial distribution is natural for the three damage counts conditional on the row total, but base R's generalized linear model interface does not supply a family named multinomial. Furthermore, the written response is one cell count, not a three-category response or grouped multinomial vector. A properly specified multinomial logistic regression could model category probabilities by group, but it is not the model fit by this command. The independent-count log-linear model in part (a), conditional on the margins, is the appropriate supplied alternative.

Under independent assignment,The arm-count vector has a multinomial distribution. Conditional on $N_r=n_r$ for $r=0,1,2$, every compatible vector has the same factor ${\prod }_r{\pi }_r^{n_r}$, sofor compatible $z$, and zero otherwise. Conditioning independent assignment on its arm sizes therefore recovers complete randomization.

For each group $g$, let independent cell counts have Poisson means ${\mu }_{gj}$. Conditional on their total $n_g$, the cells have a multinomial distribution with probabilitiesIncluding a free group main effect in the Poisson log-linear model makes its profiled likelihood for the remaining parameters proportional to the corresponding multinomial likelihood. The two fits therefore give the same fitted proportions and likelihood-ratio comparisons.

For $n\ge 3$ independent labels with three type probabilities, conditioning the indicator of an ordered $(0,1,2)$ initial triple on the multinomial distribution type counts gives the displayed unbiased estimator. The Rao-Blackwell theorem reduces its variance strictly when all three probabilities are positive. At a boundary parameter with a missing type, both variances vanish.

Place $n$ labelled balls independently into $m$ labelled bins, each with probability $1/m$. The vector of bin counts has a multinomial distribution, and each count separately has the binomial distribution $\mathrm{Bin}(n,1/m)$. Distinct bin counts have covariance $-n/m^2$, so they are dependent for $n>0$ and $m>1$. Counts of bins of a specified size are sums of indicator variables; their expectations follow from the linearity of expectation, without any independence between bins.