The exponential family can be described by \begin{align*} f(y|\theta,\phi) = \exp\left\{\phi^{-1}(y\theta - b(\theta) + c(y,\phi)\right\} \end{align*}

It also can be shown that $\textbf{E}(Y) = \mu = b^{\prime}(\theta)$ and $\textbf{Var}(Y) = \phi\ b^{\prime\prime}(\theta)$. Based on this, we can define the variance function as $V(\mu) = b^{\prime\prime}(\theta)$. My question is: based on $V$, how do we recover $\theta$? The text which I am studying says that \begin{align*} \theta = \int V^{-1}(\mu)\mathrm{d}\mu \end{align*}

Although quite simple, I am having troubles in understanding this relation. Any help is appreciated. Thanks in advance!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.