\section*{Exercise 1C (Exercise 2 p.182 in \cite{davison_2003})}

\textit{What natural exponential families are generated by (a) $f_0(y) = e^{-y}, y > 0$ and (b) $f_0(y) = \frac{1}{2}e^{-|y|}$?}

\section*{Exercise 2C (Exercise 9 p.220 in \cite{davison_2003})}

\textit{Show that the model in Example 4.5 is an exponential family. Is it steep? What happens when $R_j = 0$ whenever $x_j < a$ and $R_j = m_j$ otherwise? Find its minimal representation when all the $x_j$ are equal.}

\section*{Exercise 3C (Problems 5.11 and 5.12 in \cite{pace_salvan_1997})}

\subsection*{Exercise 5.11}

\textit{Check that the variance function associated with the negative binomial distribution is quadratic.}

\subsection*{Exercise 5.12}

\textit{Determine the natural exponential family with variance function}
\begin{align*}
    (M, V(\mu)) = ((-\infty, 0), \mu^2).
\end{align*}

\section*{Exercise 4C}

\textit{Let $Y$ be a one-dimensional rv with density belonging to an $\mathcal{F}^1_{ne}$. Let $Z \sim Y | Y < c$, where $c \in \text{int}\mathcal{Y}$ is a given constant. Show that the statistical model for $Z$ is still an $\mathcal{F}^1_{ne}$. Determine the cfg of $Z$.}

\section*{Exercise 5C (Problem 5.39 in \cite{pace_salvan_1997})}

\textit{Let $Y = (Y_1, Y_2, Y_3)$ with $Y_1 + Y_2 + Y_3 = n$, be distributed according to a trinomial distribution with index $n$ and parameter $(\pi_1, \pi_2)$. This is an element of the $\mathcal{F}^2_{e}$ with density}
\begin{align*}
    p(y; \pi_1, \pi_2) = \frac{n|}{y_1! y_2! y_3!}\pi_1^{y_1} \pi_2^{y_2} (1-\pi_1-\pi_2)^{y_3}
\end{align*}
\textit{where $\pi_1, \pi_2 > 0$ and $\pi_1 + \pi_2 < 1$. Identify the natural observations and the natural parameters. The Hardy-Weinberg model is defined as a subfamily of such an $\mathcal{F}^2_e$ obtained with the constraints $\pi_1 = \phi^2$, $\pi_2 = 2\phi(1-\phi)$, with $\phi \in (0, 1)$. Show that the constraint introduced into the natural parameter space is linear and consequently the corresponding subfamily, written in the minimal form, is an $\mathcal{F}^1_e$.}

\section*{Exercise 6C (Problem 5.24 in \cite{pace_salvan_1997})}

\textit{Obtain the representation (5.45) of the log-likelihood under random sampling from each of the exponential families described in Table 5.1.}

\section*{Exercise 7C (Problem 5.29 in \cite{pace_salvan_1997}) \\ \small (Binomial and IG)}

\textit{Obtain the uniformly most powerful unbiased test for problems (5.57) and (5.60) under random sampling of size n for each of the exponential families in Table 5.1. In particular, for $\mathsf{Ga}(\nu, \phi)$ obtain both the test on $\nu$ with $\phi$ known and that on $\phi$ with $\nu$ known. Proceed in the same way with $\mathsf{IG}(\phi, \lambda)$. For each distribution, provide some indication on finding the exact null distribution (for the inverse Gaussian, see \cite{seshadri_1993}).}

\section*{Exercise 8C (Problem 5.33 in \cite{pace_salvan_1997})}

\textit{Under the same assumptions and notation as in Example 5.12, obtain the uniformly most powerful level $\alpha$ similar test for equality of two binomial distributions with the same index, that is, for}
\begin{align*}
    H_0 : \phi_1 = \phi_2 \quad \textit{versus} \quad H_1 : \phi_1 \neq \phi_2.
\end{align*}

\section*{Exercise 9C (Exercise 3 Ch.5 in \cite{cox_hinkley_1974})}

\textit{There are available $m$ independent normal-theory estimates of variance, each with $d$ degrees of freedom. The corresponding parameters are $\sigma^2_1, \dots, \sigma^2_m$. If $\sigma^2_j = 1/(\lambda + \psi j)$, $j=1, \dots, m$, obtain a uniformly most powerful similar test of $\psi = 0$ versus alternatives $\psi > 0$. Suggest a simpler, although less efficient, procedure based on the log transformation of the variances.}

\section*{Exercise 10C (Exercise 3 Ch.5 in \cite{cox_hinkley_1974})}
Again?

\section*{Exercise 11C (Problem 6.17 in \cite{pace_salvan_1997})}

\textit{Write the contribution of a single observation to the deviance (6.57) for the generalized linear models with the canonical links summarized in Table 6.1.}