The Bernoulli identity-link boundary leaves the regular family

Given: The Bernoulli cumulant $b(\theta )=\log (1+e^{\theta })$, $x_1=1$, identity link $\eta =\mu =\beta$, and the closure boundary $\mu \downarrow 0$.

Generalized linear model with explicit conditioning: ${\mu }_i=g^{-1}({\eta }_i)$ and ${\theta }_i=\tau ({\mu }_i)$, where $\tau$ satisfies $b^{\mathrm{\prime }}(\tau (\mu ))=\mu$.

General-link score and Fisher information: $w={\dot{\mu }}^2\mathrm{/}(a(\varphi )V(\mu ))$ where $V=b^{\mathrm{\prime }\mathrm{\prime }}(\theta )$.

One minimum-norm IRLS step: the scoring step requires finite positive weights and a finite nonzero link derivative at the current iterate.

In this Bernoulli GLM, $Y_i\in \{0,1\}$ and $p(\theta )=e^{\theta }\mathrm{/}(1+e^{\theta })$; relative to counting measure the mass is $f(y;\theta )=\exp \{y\theta -\log (1+e^{\theta })\}$. It is normalized by summing over $y=0,1$. Writing the natural-parameter derivatives as $f^{\mathrm{\prime }}=f(y-p)$ and $f^{\mathrm{\prime }\mathrm{\prime }}=f\{(y-p{)}^2-p(1-p)\}$, the first two derivatives have constant integrable bounds on the two-point support, and $\mathbb{E}[Y^2]\le 1$. Thus the fixed-support, finite-moment, derivative-envelope, and score-moment hypotheses used above hold. (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials, The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t)