3.2 Equivariant Estimators
A model \(\{f_{\theta }(x):\, \theta \in (-\infty ,\infty )\}\) such that \(f_{\theta }(x) = f_0(x - \theta )\) with \(f_0\) known is called a location invariant family and \(\theta \) is called a location parameter.
Example 3.2.1. Show that the following families are location invariant:
- (a)
- \(N(\theta ,1)\)
- (b)
- \(CAU(1,\theta )\)
- (c)
- \(EXP(1,\theta )\)
Solution.
- (a)
- \(f_{\theta }(x) = \dfrac {1}{\sqrt {2\pi }}e^{-\frac {1}{2}(x - \theta )^2}\, = f_{0}(x - \theta )\, , \hspace {0.2cm} \theta \in \mathbb {R}\) where \[f_0(z) = \frac {1}{\sqrt {2\pi }}e^{-\frac {1}{2}z^2}.\]
- (b)
- \(f_{\theta }(x) = \dfrac {1}{\pi \left [1 + (x - \theta )^2\right ]} = f_0(x - \theta )\,, \hspace {0.3cm} \theta \in \mathbb {R}\) where \[f_0(z) = \frac {1}{\pi \left [1 + z^2\right ]}.\]
- (c)
- \(f_{\theta }(x) = e^{-(x - \theta )} = f_0(x - \theta )\, , \hspace {0.3cm} x > \theta \, ,\,\, x - \theta > 0\) where \[f_0(z) = e^{-z}\, , \hspace {0.3cm} z>0.\]
Definition 3.2.2. An estimator \(\tilde {\theta }(X_1, \, \cdots \, , \, X_n)\) is location equivariant if \[\tilde {\theta }(x_1 + a, \, \cdots \, , \, x_n + a) = \tilde {\theta }(x_1, \, \cdots \, , \, x_n) + a\] for any constant \(a\) and all values of \((x_1, \, \cdots \, , \, x_n)\).
Example 3.2.3. Suppose \(X_1, \, \cdots \, , \, X_n\) is a random sample from a \(N(\theta ,1)\) distribution. Show that the \(UMVUE\) is a location equivariant estimator.
Solution. \(\tilde {\theta }(X_1, \, \cdots \, , \, X_n) = \overline {X}\) is the \(UMVUE\) of \(\theta \) \begin {align*} \tilde {\theta }(x_1 + a, \, \cdots \, , \, x_n + a) & = \frac {1}{n}\sum ^n_{i = 1} (X_i + a)\\ & = \frac {1}{n}\left (\sum ^n_{i = 1} X_i + a\right )\\ & = \frac {1}{n}\sum ^n_{i = 1} X_i + a\\ & =\tilde {\theta }(x_1, \, \cdots \, , \, x_n) + a. \end {align*}
\(\therefore \, \overline {X}\) is a location equivariant estimator. □
Definition 3.2.4. A model \(\{f_{\theta }(x):\, \theta > 0\}\) such that \(f_{\theta }(x) = \frac {1}{\theta } \, f_1\left (\frac {x}{\theta }\right )\) with \(f_1\) known is called a scale invariant family and \(\theta \) is called a scale parameter.
Definition 3.2.5. An estimator \(\tilde {\theta }^k = \tilde {\theta }^k(x_1, \, \cdots \,, \, x_n)\) is scale invariant if \[\tilde {\theta }^k(cx_1, \, \cdots \,, \, cx_n) = c^k\, \tilde {\theta }^k(x_1, \, \cdots \,, \, x_n)\] for all values of \((x_1, \, \cdots \, , \, x_n)\) and \(c >0\).
Example 3.2.6. Let \(X_1, \, \cdots \, , \, X_n\) be a random sample from the \(N(0, \sigma ^2)\) distribution. Show that
- (a)
- \(f_{\theta }(x)\) is a scale invariant family
- (b)
- the \(UMVUE\) of \(\theta ^2\) is a scale equivariant estimator.
Solution.
- (a).
- \(\begin {aligned}[t] f_{\theta }(x) = \frac {1}{\sqrt {2\pi \theta ^2}}e^{-\frac {1}{2}x^2} = \frac {1}{\theta \, \sqrt {2\pi }}e^{-\frac {1}{2}\left (\frac {x}{\theta }\right )^2} = \frac {1}{\theta }\, f_1\left (\frac {x}{\theta }\right ). \end {aligned}\)
where \(f_1(z) = \dfrac {1}{\sqrt {2\pi }}e^{-\frac {1}{2}z^2}\).
\(\therefore \,\, f_{\theta }(x)\,\) is a scale invariant family.
- (b).
- \(f_{\theta }(x)\) is REF distribution with complete sufficient statistic \(T = \sum ^n_{i = 1} X_i^2.\) Thus \(\dfrac {1}{n}\sum _{i = 1}^n X_i^2\,\) is the \(UMVUE\) of \(\theta ^2\).
\[\tilde {\theta }^2(x_1, \, \cdots \,, \, x_n) = \frac {1}{n}\sum ^n_{i = 1} x_i^2.\]
\begin {align*} \tilde {\theta }^2(cx_1, \, \cdots \,, \, cx_n) & = \frac {1}{n}\sum ^n_{i = 1} \left (cx_i\right )^2 = \frac {1}{n}\sum ^n_{i = 1} c^2\, x_i^2\\ & = c^2\, \frac {1}{n}\sum ^n_{i = 1} x^2_i\\ & = c^2\, \tilde {\theta }^2(x_1, \, \cdots \,, \, x_n). \end {align*}
Therefore the \(UMVUE\) of \(\theta ^2\) is a scale equivariant estimator.
Problem 3.2.1. Show that the \(UNIF(\theta - 1, \theta + 1)\) distribution is a location invariant family.
Show solution
Solution. The UNIF\((\theta -1,\theta +1)\) density is \[f_{\theta }(x) = \frac 12\,I\left (\theta -1<x<\theta +1\right ) = \frac 12\,I\left (-1<x-\theta <1\right ) = f_0(x-\theta ),\] where \(f_0\) is the UNIF\((-1,1)\) density. Since every member of the family is obtained from the single density \(f_0\) by a shift, the family is location invariant with location parameter \(\theta \).
Equivalently: if \(X\sim \) UNIF\((\theta -1,\theta +1)\) then \(X+c\sim \) UNIF\((\theta +c-1,\ \theta +c+1)\), so translating the data by \(c\) moves the parameter by \(c\) and leaves the family as a whole unchanged.
Problem 3.2.2. Let \(X_1, \, \cdots \, , \, X_n\) be a random sample the \(UNIF(0,\theta )\) distribution. Show that then
- (a)
- \(UNIF(0,\theta )\) densities is a scale invariant family.
- (b)
- \(UMVUE\) of \(\theta \) is a scale equivariant estimator.
Show solution
Solution.
(a)
The UNIF\((0,\theta )\) density is \[f_{\theta }(x) = \frac {1}{\theta }\,I\left (0<x<\theta \right ) = \frac {1}{\theta }\,I\left (0<\frac {x}{\theta }<1\right ) = \frac {1}{\theta }f_1\!\left (\frac {x}{\theta }\right ),\] with \(f_1\) the UNIF\((0,1)\) density. Every member is obtained from \(f_1\) by the scaling \(x\mapsto x/\theta \), so the family is scale invariant. Equivalently, if \(X\sim \) UNIF\((0,\theta )\) and \(c>0\) then \(cX\sim \) UNIF\((0,c\theta )\).
(b)
Since \(E\left (X_{(n)}\right )=n\theta /(n+1)\) and \(X_{(n)}\) is complete and sufficient, the UMVUE is \[\widehat {\theta } = \frac {n+1}{n}X_{(n)} .\] Scale equivariance asks that \(\widehat {\theta }(cX_1,\dots ,cX_n) = c\,\widehat {\theta }(X_1,\dots ,X_n)\) for every \(c>0\). The maximum of the rescaled sample is \(c\,X_{(n)}\), so \[\widehat {\theta }(cX) = \frac {n+1}{n}\,cX_{(n)} = c\,\widehat {\theta }(X),\] and the estimator is scale equivariant.
Note. Equivariance is a consistency requirement, not an optimality one: it says only that measuring the data in different units should change the estimate in the same units. It is worth checking because an estimator that fails it is answering a different question depending on whether the lifetimes were recorded in days or in hours.
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