The Principle of Equivariance

Show that the class $G(\mathcal{C})$ is a group.

In Example 2.2(ii), show that the transformations $\mathbf{x}^{\prime }=-\mathbf{x}$ together with the identity transformation form a group.

Let $\left\{ g X, g \in G\right\}$ be a group of transformations that leave the model (2.1) invariant. If the distributions $P_{\theta }, \theta \in \Omega$ are distinct, show that the induced transformations $\bar{g}$ are $1: 1$ transformations of $\Omega$.

Under the assumptions of Problem 2.3, show that

Show that a loss function satisfies (2.9) if and only if it is of the form (2.10).

Let $X$ be distributed as $N\left(\xi , \sigma^{2}\right),-\infty <\xi <\infty , 0<\sigma$, and let $h(\xi , \sigma )=\sigma^{2}$. The problem is invariant under the transformations $x^{\prime }=a x+c ; 0<a,-\infty <c<\infty$. Show that the only equivariant estimator is $\delta (X) \equiv 0$.

Show that:

If $\theta$ is the true temperature in degrees Celsius, then $\theta^{\prime }=\bar{g} \theta =\theta +273$ is the true temperature in degrees Kelvin. Given an observation $X$, in degrees Celsius:

To illustrate the difference between functional equivariance and formal invariance, consider the following. To estimate the amount of electric power obtainable from a stream, one could use the estimate

where $x=$ stream flow in $\mathrm{m}^{3} / \mathrm{sec}, 100 \mathrm{~ m}^{3} / \mathrm{sec}$ is the capacity of the pipe leading to the turbine, and $20 \mathrm{~ m}^{3} / \mathrm{sec}$ is the flow reduction necessary to avoid harming the trout. The constant $c$, in kilowatts $/ \mathrm{m}^{3} / \mathrm{sec}$ converts the flow to a kilowatt estimate.

In an invariant probability model, write $X=(T, W)$, where $T$ is sufficient for $\theta$, and $W$ is ancillary.

In an invariant estimation problem, write $X=(T, W)$ where $T$ is sufficient for $\theta$, and $W$ is ancillary. If the group of transformations is transitive, show:

For the situation of Example 2.11:

For the situation of Example 2.12:

Theorem 2.7 states: if $\delta$ is an equivariant estimator in a problem which is invariant under a transformation group $G$, then the risk function of $\delta$ satisfies $R(\bar{g}\theta ,\delta )=R(\theta ,\delta )$ for all $\theta$ and all $g\in G$.

Corollary 2.13 states: under the assumptions of Theorem 2.7, the risk function of any equivariant estimator is constant on the orbits of $G$.

Prove Corollary 2.13.

If $\delta (X)$ is an equivariant estimator of $h(\theta )$ under a group $G$, then so is $g^{*} \delta (X)$ with $g^{*}$ defined by (2.12) and (2.13), provided $G^{*}$ is commutative.

Show that:

In Example 2.14, determine the totality of equivariant estimators of $\Delta$ under the smallest group $G$ containing $G_{1}$ and $G_{2}$.

Let $\theta$ be real-valued and $h$ strictly increasing, so that (2.11) is vacuously satisfied. If $L(\theta , d)$ is the loss resulting from estimating $\theta$ by $d$, suppose that the loss resulting from estimating $\theta^{\prime }=h(\theta )$ by $d^{\prime }=h(d)$ is $M\left(\theta^{\prime }, d^{\prime }\right)=L\left[\theta , h^{-1}\left(d^{\prime }\right)\right]$. Show that:

If $\delta (\mathbf{X})$ is MRE for estimating $\xi$ in Example 2.2(i) with loss function $\rho (d-\xi )$, state an optimum property of $e^{\delta (\mathbf{X})}$ as an estimator of $e^{\xi }$.

Let $X_{i j}, j=1, \ldots , n_{i}, i=1, \ldots , s$, and $W$ be distributed according to a density of the form

where $\mathbf{x}_{i}-\xi_{i}=\left(x_{i 1}-\xi_{i}, \ldots , x_{i n_{i}}-\xi_{i}\right)$, and consider the problem of estimating $\theta =\Sigma c_{i} \xi_{i}$ with loss function $L\left(\xi_{i}, \ldots , \xi_{s} ; d\right)=\rho (d-\theta )$. Show that:

Theorem 1.4 states: let $X$ be distributed with density (1.9), and let $\delta$ be equivariant for estimating $\xi$ with loss function (1.15). Then, the bias, risk, and variance of $\delta$ are all constant (i.e., do not depend on $\xi$).

Generalize Theorem 1.4 to the situation of Problem 2.23.

If $\delta_{0}$ is any equivariant estimator of $\theta$ in Problem 2.23, and if $\mathbf{y}_{i}=\left(x_{i 1}-x_{i n_{i}}, x_{i 2}-x_{i n_{i}}, \ldots , x_{i n_{i}-1}-x_{i n_{i}}\right)$, show that the most general equivariant estimator of $\theta$ is of the form

Theorem 1.10 states: let $X=(X_{1}, \ldots , X_{n})$ be distributed according to (1.9), let $Y_{i}=X_{i}-X_{n}$ $(i=1, \ldots , n-1)$ and $Y=(Y_{1}, \ldots , Y_{n-1})$. Suppose that the loss function is given by (1.15) and that there exists an equivariant estimator $\delta_{0}$ of $\xi$ with finite risk. Assume that for each $y$ there exists a number $v(y)=v^{*}(y)$ which minimizes $\mathbb {E}_{0}\left[\rho [\delta_{0}(X)-v(y)] \mid y\right]$. Then, a location equivariant estimator $\delta$ of $\xi$ with minimum risk exists and is given by $\delta^{*}(X)=\delta_{0}(X)-v^{*}(Y)$.

Corollary 1.12 states: under the assumptions of Theorem 1.10: (i) if $\rho (d-\xi )=(d-\xi )^{2}$, then $v^{*}(y)=\mathbb {E}_{0}\left[\delta_{0}(X) \mid y\right]$; (ii) if $\rho (d-\xi )=\left|d-\xi \right|$, then $v^{*}(y)$ is any median of $\delta_{0}(X)$ under the conditional distribution of $X$ given $y$.

Suppose that the variables $X_{i j}$ in Problem 2.23 are independently distributed as $N\left(\xi_{i}, \sigma^{2}\right), \sigma$ is known. Show that:

Show that the transformation of Example 2.11 and the identity transformation are the only transformations leaving the family of binomial distributions invariant.