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For fixed known $r>0$, take the natural parameter of an exponential familyThe negative binomial distribution has log-likelihoodThus this is a negative binomial exponential family, with dispersion parameter $\varphi =1$ in the convention of part (a). The carrier contains the Gamma function and depends on the fixed size $r$, which need not be an integer. Differentiating the cumulant function of an exponential family givesThereforeThe size $r$ controls overdispersion relative to a Poisson distribution; it is different from the unit dispersion parameter used by this fixed-size generalized linear model. Another common convention applies to the scaled response $Z=Y/r$: then $b(\theta )=-\log (1-e^{\theta })$ and $\varphi =1/r$. Keeping track of the response scale prevents confusing these two representations. At $\lambda =0$ the law is concentrated at zero and is obtained as a boundary limit, rather than by a finite natural parameter of an exponential family.

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.

This does not model the full three-category response. A binomial distribution has two outcome classes, whereas the observed damage factor has three. In base R, a factor response to a binomial generalized linear model is coded as first level versus all other levels; this would silently collapse the categories. It would also treat the nine table rows as nine individual observations, ignoring the frequencies. A deliberate binary analysis would require specifying the collapse and retaining the counts, but cannot recover the full damage distribution.

Fit a Poisson regression to artificial responses $Z_i=1$ with meansWith the logarithmic link, this is the generalized linear modelwhere $\log \Lambda (Y_i)$ is a known offset. Its log likelihood, apart from constants, isThe differencedoes not depend on $\beta$. The Poisson surrogate therefore has exactly the same maximum-likelihood estimate of $\beta$. This is the Poisson surrogate for an uncensored proportional-hazards likelihood.

For independent observations from an exponential dispersion family with common known dispersion parameter ${\sigma }^2$, the generalized linear model log-likelihood iswhere ${\mu }_i=b^{\prime }({\theta }_i)$ and the link function satisfies $g({\mu }_i)={\eta }_i=x_i^T\beta$.

A suitable generalized linear model is a Gamma regression with logarithmic link:withThe gamma distribution is appropriate for a positive, continuous, right-skewed payment, and its variance function allows the conditional variance to grow like the square of the conditional mean instead of assuming constant variance. The logarithmic link function also guarantees positive fitted means and represents covariate effects multiplicatively, which accords with the Box–Cox transformation evidence for a logarithmic scale.

At a current value $\beta$, putand define the working responseThen the displayed formulas in the question can be written compactly asandIndeed, the $j$th component of the latter iswhich is precisely the stated score.

Substitution into the Fisher-scoring update givesBy part (b), this is the weighted least squares fit of the working response $z$ on $X$. Both $z$ and $W$ are recomputed from the current fitted means after every update. Fisher scoring for a generalized linear model is therefore the iteratively reweighted least squares algorithm.

A generalized linear model assumes independent responses from one exponential dispersion family,where ${\sigma }_i^2=a_i{\sigma }^2$, ${\mu }_i=\mathbb{E}[Y_i]=b^{\prime }({\theta }_i)$, and a link function relates the mean to the linear predictor by $g(\mu )=X\beta$. It follows from normalization of the density after shifting the natural parameter thatIndependence therefore gives the joint moment-generating functionwhenever every shifted natural parameter lies in its natural-parameter domain.

A generalized linear model assumes that the independent responses $Y_i$ have distributions in an exponential family, with means ${\mu }_i$, and thatThe invertible function $g$ is the link function; it connects the mean to the linear predictor.

For binomial regression, write $Y_i\sim \mathrm{Bin}(m_i,p_i)$ independently, so ${\mu }_i=m_i{p}_i$. The logistic regression link iswhile probit regression useswhere $\Phi$ is the standard-normal distribution function. The logit is the canonical link function for the binomial family.

The ordinary least squares equations for model 2 areFor the Gamma generalized linear model, $V(\mu )={\mu }^2$ and $d\mu /d\eta =\mu$, so its score equation under the logarithmic link isWhen $T_i$ is close to ${\mu }_i$,by the first-order Taylor expansion of the exponential function. The Gamma score equations then become the model-2 normal equations, so their coefficient estimates are close.

A generalized linear model has three components:

For independent uncensored event times $Y_i$ with $h_i(t)=\lambda (t)e^{{\beta }^T{x}_i}$ and baseline cumulative hazard $\Lambda (t)={\int }_0^t\lambda (s)ds$, maximizing the survival log likelihood over $\beta$ is equivalent to fitting a Poisson regression with unit responses and meansThus a log-link generalized linear model with response $1$, linear predictor ${\beta }^T{x}_i$, and offset $\log \Lambda (Y_i)$ gives the same estimate of $\beta$.