\sum_{i=1}^n \psi_i(\theta).$$ Note that $\psi_i(\theta)$ is the cumulant generating function (potential function) for the logistic regression model, which is an exponential family model with the natural parameter $\theta$. Hence each $\psi_i(\theta)$ and $\psi(\theta)$ are convex in $\theta$. Indeed by , the Hessian matrix $H_{\psi_i}(\theta)$ of $\psi_i$ is given as $$H_{\psi_i}(\theta)=
c_i c_i{'}\frac{p_i (1-p_i)e^{{\theta{'}}c_i}}{(1+ p_i (e^{{\theta{'}}c_i}-1))^2},$$ which is non-negative definite. The Hessian matrix $$H_{\psi}(\theta)=\sum_{i=1}^n H_{\psi_i}(\theta)$$ of $\psi$ is positive definite if ${{\cal V}}_n$ is positive definite, which is the Fisher information matrix in terms of the natural parameter $\theta$.

Convexity of $\psi_i$ implies concavity of $\log {{\cal K}}_n^\theta={\theta{'}}{{\cal T}}_n - \psi(\theta)$, where $${{\cal T}}_n = \sum_{i=1}^n c_i x_i = {{\cal S}}_n + \sum_{i=1}^n c_i p_i.$$ Hence if the maximum of $\log {{\cal K}}_n^\theta$ is attained at a finite value ${\hat\theta_n^*}$, then the “maximum likelihood estimate” ${\hat\theta_n^*}$ satisfies “the likelihood equation” $$\frac{\partial}{\partial \theta} \log {{\cal K}}_n^\theta=0$$ or equivalently $$\label{eq:likelihood-equation2}
{{\cal T}}_n = \nabla \psi({\hat\theta_n^*}).$$ The likelihood equation can also be written as $$0=\sum_{i=1}^n (x_i - \hat p_i^*)c_i,  \qquad
\hat p_i^*  = \hat p_{i;n}^*
= \frac{p_i e^{{\hat\theta_n^*}c_i}}{1+p_i (e^{{\hat\theta_n^*}c_i}-1)}.$$ From this it follows that ${\hat\theta_n^*}=0$ if and only if ${{\cal T}}_n = \sum_{i=1}^n c_i p_i$.

Regard as determining ${\hat\theta_n^*}$ in terms of $t={{\cal T}}_n$, 