= Solution
Let $X$ be the $n\times p$ <design matrix> with row $x_i^T$, let $y=(y_i)$, and write $\mu_i(\beta)=\exp(x_i^T\beta)$. Independence gives the <Poisson regression> <likelihood function>
$$
L(\beta)=\prod_{i=1}^n\frac{e^{-\mu_i}\mu_i^{y_i}}{y_i!},\qquad
\ell(\beta)=\sum_i\{y_ix_i^T\beta-e^{x_i^T\beta}-\log(y_i!)\}.
$$
The <score function> and negative <Hessian matrix> are
$$
\boxed{U(\beta)=X^T(y-\mu),\qquad I(\beta)=-\nabla^2\ell(\beta)=X^TWX,\quad W=\operatorname{diag}(\mu_i).}
$$
Because $EY_i=\mu_i$, the score has mean zero; independence and the <Poisson distribution> <variance> give $\operatorname{Var}U=X^TWX$. Thus the observed and expected <Fisher information matrices> are the same here.
Assume $X$ has full column <matrix rank> $p$. For a nonzero vector $a$, $a^TIa=\sum_i\mu_i(x_i^Ta)^2>0$. The <concavity of the Poisson regression likelihood> therefore makes any finite solution of $X^T(y-\widehat\mu)=0$ the unique <maximum-likelihood estimator>. If the design is rank deficient, parameters differing by a vector in $\ker X$ give the same distribution, so only identifiable combinations can be estimated. Full rank alone does not ensure a finite maximizer: for an intercept-only model with all observations zero, $\ell=-ne^\beta$ approaches its supremum as $\beta\to-\infty$. More generally, directions that decrease only fitted means attached to zero counts can produce boundary fits. A useful sufficient case for a finite fit is full rank with every count positive: each summand $y_i\eta_i-e^{\eta_i}$ tends to $-\infty$ in either tail, and some $|\eta_i|$ diverges whenever $\|\beta\|$ diverges.
For an interior fit, <Newton's method> and <Fisher scoring> give the same update:
$$
\beta^{(k+1)}=\beta^{(k)}+[X^TW^{(k)}X]^{-1}X^T(y-\mu^{(k)}).
$$
This is also <iteratively reweighted least squares>. Define the working response $z_i=\eta_i+(y_i-\mu_i)/\mu_i$, with $\eta=X\beta^{(k)}$, and regress $z$ on $X$ using weights $W^{(k)}$:
$$
\beta^{(k+1)}=(X^TW^{(k)}X)^{-1}X^TW^{(k)}z.
$$
A numerical implementation solves these linear equations rather than explicitly forming the inverse, checks convergence of the score and <likelihood>, and can shorten a step when necessary.
Under the usual fixed-design regularity conditions, with growing information and no single observation dominating it, a score <Taylor expansion> about the true parameter gives
$$
\widehat\beta-\beta_0\approx I(\beta_0)^{-1}U(\beta_0),\qquad
\boxed{\widehat\beta\ \dot\sim\ N_p\big(\beta_0,I(\beta_0)^{-1}\big).}
$$
Estimate the <covariance matrix> by $[X^T\operatorname{diag}(\widehat\mu_i)X]^{-1}$; its diagonal square roots are the coefficient <standard errors>. This justifies large-sample <Wald confidence intervals> and <Wald tests> for specified coefficients or contrasts. Finite-sample unbiasedness is not being asserted. A <likelihood-ratio test> compares nested fits through twice their maximized <log-likelihood> difference. The <Poisson deviance> against the saturated fit is
$$
D=2\sum_i\left\{y_i\log\frac{y_i}{\widehat\mu_i}-(y_i-\widehat\mu_i)\right\},
$$
with $0\log0=0$. Residual diagnostics and a check of <overdispersion> assess whether the assumed Poisson <variance> is adequate; the <variance> correction required when it is not is derived in the next part.
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