= Solution
An <exponential dispersion family of order one> has density or mass function, relative to a fixed dominating measure, of the form
$$
p(y;\theta,\phi)=\exp\!\left\{\frac{y\theta-b(\theta)}{\phi}+c(y,\phi)\right\},
$$
where $\theta$ is a scalar <natural parameter>, $\phi>0$ is the <dispersion parameter>, and the support does not depend on $\theta$. For regular families, $b$ is twice differentiable on an open natural-parameter domain and $b''>0$. Known observation weights $w$ replace $\phi$ by $\phi/w$. “Order one” refers to the scalar canonical <statistic> $y$, rather than an arbitrary vector of canonical <statistics>.
Differentiate the normalization $\int p(y;\theta,\phi)\,dy=1$, or its discrete counterpart, assuming differentiation under the integral is valid. The first <derivative> gives $\mathbb E(Y-b'(\theta))=0$. The second <derivative> gives
$$
0=\mathbb E\!\left[\frac{(Y-b'(\theta))^2}{\phi^2}-\frac{b''(\theta)}\phi\right].
$$
Consequently,
$$
\boxed{\mu=\mathbb E Y=b'(\theta),\qquad \operatorname{Var}(Y)=\phi b''(\theta).}
$$
The <variance function> expresses the second <derivative> in terms of the mean:
$$
\boxed{V(\mu)=b''((b')^{-1}(\mu)),\qquad \operatorname{Var}(Y)=\phi V(\mu)/w.}
$$
A <generalized linear model> specifies independent responses from an <exponential dispersion family>, a <linear predictor> $\eta_i=x_i^T\beta$, and a differentiable monotone <link function> relating it to the response mean by $g(\mu_i)=\eta_i$. The family, <variance function>, covariate design, <link function> and dispersion specification together define the model. The <canonical link function> identifies the <linear predictor> with the <natural parameter>:
$$
\boxed{g(\mu)=(b')^{-1}(\mu),\qquad\theta_i=x_i^T\beta.}
$$
One advantage is that the <likelihood> equations become particularly simple. With known weights $w_i$ and common dispersion $\phi$, the <score function> and negative <Hessian matrix> for $\beta$ are
$$
U_\beta=\frac1\phi\sum_i w_ix_i(Y_i-\mu_i),\qquad
-\ell_{\beta\beta}=\frac1\phi\sum_iw_iV(\mu_i)x_ix_i^T.
$$
The latter does not explicitly depend on the responses, so it equals the expected <Fisher information matrix> at the same parameter. With full-rank design and positive <variances> the <likelihood> is strictly concave in $\beta$, simplifying optimization. These statements concern the coefficient block at a fixed dispersion value.
Three explicit examples, taking unit weights, are as follows.
For a <normal distribution> with mean $\mu$ and <variance> $\phi$,
$$
\log p(y)=\frac{y\mu-\mu^2/2}{\phi}-\frac{y^2}{2\phi}-\frac12\log(2\pi\phi).
$$
Thus $\theta=\mu$, $b(\theta)=\theta^2/2$, $c(y,\phi)=-y^2/(2\phi)-\tfrac12\log(2\pi\phi)$, and $V(\mu)=1$. Its <canonical link function> is $\boxed{g(\mu)=\mu}$, the <identity link>.
For a <Poisson distribution> with mean $\mu>0$ and support $\{0,1,\ldots\}$,
$$
\log p(y)=y\log\mu-\mu-\log(y!).
$$
Here $\phi=1$, $\theta=\log\mu$, $b(\theta)=e^\theta$, and $c(y,1)=-\log(y!)$. Therefore $b'(\theta)=b''(\theta)=e^\theta$, $V(\mu)=\mu$, and the <Poisson canonical link> is $\boxed{g(\mu)=\log\mu}$.
For a <Bernoulli distribution> with success probability $\mu\in(0,1)$,
$$
\log p(y)=y\log\frac\mu{1-\mu}+\log(1-\mu),\qquad y\in\{0,1\}.
$$
Taking $\phi=1$, $\theta=\log(\mu/(1-\mu))$, $b(\theta)=\log(1+e^\theta)$, and $c(y,1)=0$ gives the required form. Its mean and <variance function> are $b'(\theta)=e^\theta/(1+e^\theta)=\mu$ and $V(\mu)=\mu(1-\mu)$. Its <canonical link function> is $\boxed{g(\mu)=\log(\mu/(1-\mu))}$, the <logit link>. The fixed dispersion values in the Poisson and Bernoulli cases are part of those models; an arbitrary continuous dispersion is not being asserted for them.
To test $H_0:\beta_j=0$, first fit the unrestricted model and then the restricted model with that component fixed at zero, estimating all <nuisance parameters>, including dispersion when applicable, in each fit. If $\widehat\vartheta$ and $\widetilde\vartheta$ denote their full fitted parameter vectors, the <likelihood-ratio test> uses
$$
\boxed{D=2\{\ell(\widehat\vartheta)-\ell(\widetilde\vartheta)\}.}
$$
For identifiable regular models, an interior null parameter and a fixed-dimensional full-rank design with increasing information, the <Wilks theorem> gives $D\xrightarrow{d}\chi_1^2$ under $H_0$. A level-$\alpha$ test rejects for $D$ greater than the $(1-\alpha)$-<quantile> of the <chi-squared distribution> with one <degree of freedom>.
Alternatively the <Wald test> uses $W=\widehat\beta_j^2/\widehat{\operatorname{Var}}(\widehat\beta_j)$, with the <variance> obtained from the appropriate inverse full <Fisher information matrix>; it has the same limiting $\chi_1^2$ null law. The <score test> fits only the restricted model. If $\lambda$ collects its <nuisance parameters>, its efficient information is $I_{jj\cdot\lambda}=I_{jj}-I_{j\lambda}I_{\lambda\lambda}^{-1}I_{\lambda j}$, and the <statistic> is $U_j(\widetilde\vartheta)^2/I_{jj\cdot\lambda}(\widetilde\vartheta)$, again asymptotically $\chi_1^2$. Accounting for the nuisance block is necessary rather than using the unadjusted diagonal information.
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