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\begin{document}

\title{Stats 210A, Fall 2025\\
  Homework 3 \\
  {\large {\bf Due on}: Wednesday, Sep. 24}}
\date{}

\maketitle

\paragraph{Instructions:} You may disregard measure-theoretic niceties about conditioning on measure-zero sets, almost-sure equality vs. actual equality, ``all functions'' vs. ``all measurable functions,'' etc. (unless the problem is explicitly asking about such issues).

\begin{problem}[Multinomial subfamilies]
  \label{prob:multinomial-subfamilies}

  The multinomial family is a multi-category version of the binomial, it measures the number of times each category comes up if we sample a $d$-category random variable with distribution $\pi$ on $n$ independent trials. Throughout this problem assume $d \geq 3$.

  If $X \sim \text{Multinom}(n, \pi)$, with all $\pi_j > 0$ and $\sum_j \pi_j = 1$, then $X$ has density
  \[
  p_\pi(x) = \pi_1^{x_1}\pi_2^{x_2}\cdots \pi_d^{x_d} \cdot \frac{n!}{x_1! x_2! \cdots x_d!}
  \]
  
  {\bf Note:} The coordinates of $X=(X_1,\ldots,X_d)$ are neither independent nor identically distributed.

 \begin{enumerate}[(a)]
    
  \item Rewrite the densities as a $(d-1)$-parameter exponential family, giving an explicit form for $T(x)$, $h(x)$, $\eta$, and $A(\eta)$. Show whether $X=(X_1,\ldots,X_d)$ is complete sufficient, minimal sufficient, or neither.

    
  
  \item Suppose a certain gene has two alleles {\bf A} and {\bf a}, and $\theta\in (0,1)$ is the unknown prevalence of allele {\bf a} in a well-mixed population. Then the proportion of people in the population with genotypes {\bf aa}, {\bf Aa}, and {\bf AA} is $\theta^2$, $2\theta(1-\theta)$, and $(1-\theta)^2$, respectively.

    We can estimate $\theta$ by sampling $n$ independent individuals from the population and counting the number who have each genotype. These counts will have a joint multinomial distribution with probability parameter
    \[
    \pi(\theta) = (\theta^2, 2\theta(1-\theta), (1-\theta)^2).
    \]
    Hence, scientific considerations might lead us to use the multinomial subfamily indexed by $\theta$:
    \[
    \cP = \{\text{Multinom}(n,\pi(\theta)):\; \theta \in (0,1)\}.
    \]
    Can $\cP$ be written as a one-parameter exponential family? Find a minimal sufficient statistic for $\cP$, and show whether or not it is complete.

    


  \item Now suppose our population is a mixture of two populations with different prevalences $\theta_1$ and $\theta_2$ for allele {\bf a}. Define $\gamma \in (0,1)$ as the proportion of individuals from population 1. Assume that $\theta_1,\theta_2$ are known and only $\gamma$ is unknown. Since $\theta_1$ and $\theta_2$ are known it may be convenient to write the mixture probabilities as
    \[
    \pi(\gamma) = \gamma\pi^{(1)} + (1-\gamma)\pi^{(2)}, \quad \text{ for } \pi^{(k)} = (\theta_k^2, 2\theta_k(1-\theta_k), (1-\theta_k)^2), \;\;k=1,2.
    \]

    Now suppose that we again sample $n$ individuals from our unknown mixture, giving another one-parameter subfamily $\cQ$ indexed by $\gamma$. Can $\cQ$ be written as a one-parameter exponential family? Find a minimal sufficient statistic for $\cQ$, and show whether or not it is complete.

  

  \end{enumerate}

  \paragraph{Moral:} The structure of the families and subfamilies determines the properties of the sufficient statistic.

\end{problem}


\begin{problem}[Gamma family]
\label{prob:gamma-family}

The gamma family is a two-parameter family of distributions on $\RR_+ = [0,\infty)$, with density
\[
p_{k,\theta}(x) = \frac{x^{k-1}e^{-x/\theta}}{\Gamma(k)\theta^k}
\]
with respect to the Lebesgue measure on $\RR_+$. $k>0$ and $\theta>0$ are respectively called the shape and scale parameters, and $\Gamma(k)$ is the gamma function, defined as
\[
\Gamma(k) = \int_0^\infty x^{k-1}e^{-x}\,d x.
\]
The gamma distribution generalizes the exponential distribution 
\[
\text{Exp}(\theta) = \theta^{-1}e^{-x/\theta} = \text{Gamma}(1,\theta)
\]
and the chi-squared distribution 
\[
\chi_d^2 = \frac{x^{d/2-1}e^{-x/2}}{\Gamma(d/2)2^{d/2}} = \text{Gamma}(d/2,2).
\]



\begin{enumerate}[(a)]
\item Show that the Gamma is a 2-parameter exponential family by putting it into its canonical form. Find the natural parameter, sufficient statistic, carrier density, and log-partition function ({\bf Note}: there are multiple valid ways of doing this).

  

\item Find the mean and variance of $X \sim \Gamma(k,\theta)$.

  

\item Find the moment generating function of $X\sim \Gamma(k,\theta)$:
\[
M_X(u) = \EE_{k,\theta}[e^{uX}],
\]
and use it to find the distribution of $X_+ = \sum_{i=1}^n X_i$ where $X_1,\ldots,X_n$ are mutually independent with $X_i \sim \text{Gamma}(k_i, \theta)$.

You may use without proof the following uniqueness result about MGFs:
If $Y$ and $Z$ are two random variables whose MGFs coincide in a neighborhood of 0 ($\exists \delta>0$ for which $M_Y(u) =M_Z(u) < \infty$ for all $u\in[-\delta,\delta]$), then $Y$ and $Z$ have the same distribution.


\end{enumerate}
\end{problem}


\begin{problem}[Interpretation of completeness]
\label{prob:interp-completeness}

The concept of {\em completeness} for a family of measures was introduced in \citet{lehmann1950completeness} as a precursor to their definition, in the same paper, of a complete statistic. The definition of a complete family did not stick, and lives on only in the (consequently confusingly named) idea of complete statistic (in particular it has nothing to do with the definition of a {\em complete measure} that you can find on Wikipedia).

If $\cP = \{P_\theta:\; \theta \in \Theta\}$ is a family of measures on $\cX$, we say that $\cP$ is {\em complete} if 
\[
\int f(x)\,d  P_\theta(x) = 0, \;\forall \theta \quad\Rightarrow\quad
P_\theta(\{x:\; f(x) \neq 0\}) = 0, \;\forall \theta.
\]
This can be interpreted as an inner product $\langle f, P_\theta\rangle = \int f\,d  P_\theta$, where $f \perp P_\theta$ if $\langle f, P_\theta\rangle = 0$. Then, the family is {\bf not} complete if there is some nonzero function $f$ that is orthogonal to every $P_\theta$. We will try to gain some intuition for this definition and, thereby, for the definition of a complete statistic.

For the following parts, let $\cP = \{P_\theta:\; \theta \in \Theta\}$ be a family of probabilty measures on $\cX$, assume $T(X)$ is a statistic, and let $\cT = T(\cX)$ be the range of the statistic $T(X)$. Let $\cP^T = \{P_\theta^T:\; \theta\in \Theta\}$ denote the induced model of push-forward probability measures on $\cT$ denoting the possible distributions of $T(X)$:
\[
P_\theta^T(B) = P_\theta(T^{-1}(B)) = \PP_\theta(T(X) \in B).
\]

\begin{enumerate}[(a)]
\item Show that $T(X)$ is a complete statistic for the family $\cP$ if and only if $\cP^T$ is a complete family.



\item Assume (for this part only) that $\cX$ is a finite set, i.e. $\cX = \{x_1,\ldots, x_n\}$ for some $n<\infty$, and assume without loss of generality that every $x\in\cX$ has $P_\theta(\{x\}) > 0$ for at least one value of $\theta$ (otherwise we could truncate the sample space). 

Let $p_\theta(x) = \PP_\theta(X = x) \geq 0$, and $v^\theta = (p_\theta(x_1),\ldots,p_\theta(x_n)) \in \RR^n$. Show that $\cP$ is complete if and only if $\text{Span}\{v^\theta:\;\theta\in\Theta\} = \RR^n$.



\item Let $X_1,\ldots,X_n \simiid \text{Pois}(\theta)$ for $\theta\in \Theta = \{\theta_1,\ldots, \theta_m\}$ with $2 \leq m < \infty$. Find a sufficient statistic that is minimal but not complete (prove both properties).

  

\item \optional In the same scenario but with $\Theta = \pi\ZZ_+ = \{0, \pi, 2\pi, \ldots\}$, show that the same statistic is minimal but not complete.

{\bf Hint:} Recall the Taylor series 
\[
\sin(\theta) = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots.
\]



\item \optional  Let $X_1,\ldots,X_n \simiid \text{Pois}(\theta)$ for $\theta\in \Theta$, and assume that $\Theta$ has an accumulation point at 0, i.e. $\Theta$ includes an infinite sequence of positive values $\theta_1,\theta_2,\ldots\in \Theta$ such that $\lim_{m\to\infty} \theta_m = 0$. Find a complete sufficient statistic and prove it is complete sufficient. 

{\bf Hint:} suppose $f$ is a counterexample function; what is $f(0)$? It may be helpful to recall that $\int f\,d  \mu$ is undefined unless either $\int \max(0,f(x))\,d  \mu(x)$ or $\int \max(0, -f(x))\,d  \mu(x)$ is finite; as a result $\int f\,d  \mu = 0 \Rightarrow \int|f|\,d  \mu < \infty$.


\end{enumerate}

\paragraph{Moral 1:} The definition of a complete statistic is easier to remember if we recall its interpretation as saying that the set of distributions $P_\theta^T$ ``spans'' a certain vector space, so that only the zero function is orthogonal to all $P_\theta^T$.

\paragraph{Moral 2:} If $\cP = \{P_\eta:\; \eta \in \Xi\}$ is a full-rank exponential family with natural parameter $\eta$, meaning $\Xi$ contains an open set, our result from class allows us to prove completeness of $T(X)$. But the converse is far from true: it is possible for $T$ to be complete if $\Xi$ is discrete, or even finite.
\end{problem}



\begin{problem}[Ancillarity in location-scale families]
\label{prob:ancillary-locscale}

In a parameterized family where $\theta = (\zeta, \lambda)$, we say a statistic $T$ is {\em ancillary for $\zeta$} if its distribution is independent of $\zeta$; that is, if $T(X)$ is ancillary in the subfamily where $\lambda$ is known, for each possible value of $\lambda$.

Suppose that $X_1,\ldots,X_n\in \cX=\RR$ are an i.i.d. sample from a {\em location-scale family} $\cP = \{F_{a,b}(x) = F((x-a)/b): \; a\in \RR, b>0\}$, where $F(\cdot)$ is a known cumulative distribution function. The real numbers $a$ and $b$ are called the {\em location} and {\em scale} parameters respectively. 

{\bf Note:} It is {\em not} enough to prove ancillarity of the coordinates; the joint distribution of the statistic shouldn't depend on the relevant parameter.
\begin{enumerate}[(a)]
\item Show that the vector of differences $\left(X_1 - X_i\right)_{i = 2}^n$ is ancillary for $a$.


\item Show that the vector of ratios $\left(\frac{X_1 - a}{X_i - a}\right)_{i=2}^n$ is ancillary for $b$. (Note: this is only a statistic when $a$ is known).


\item \optional Show that the vector of difference ratios $\left(\frac{X_1 - X_i}{X_2 - X_i}\right)_{i=3}^n$ is ancillary for $(a,b)$.

\item Let $X_1,\ldots,X_n$ be mutually independent with $X_i \sim \text{Gamma}(k_i, \theta)$. Show that $X_+ = \sum_{i=1}^n X_i$ is independent of $(X_1,\ldots,X_n)/X_+$.


\end{enumerate}

\paragraph{Moral:} Location-scale families have common structure that we can exploit to derive a wide variety of results.
\end{problem}




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