4.3 Locally Most Powerful tests
It is not always possible to construct uniformly most powerful tests. In most problems, alternative parameter values that are close to the null hypothesis are hard to detect. It is therefore important to use a test that has high a power as possible for alternative parameter values that are close to the null hypothesis. Such a test is called locally most powerful.
Definition 4.3.1. A test of \(H_0:\, \theta = \theta _0\) against \(H_1:\, \theta > \theta _0\) with power function \(\beta (\theta )\) is locally most powerful if, for any other test having the same size and power function \(\beta ^*(\theta )\), there exists an \(\varepsilon > 0\) such that \(\beta (\theta )\geq \beta ^*(\theta )\) for all \(\theta _0 < \theta < \theta _0 + \varepsilon \).
Theorem 4.3.2. Suppose \(X_1, \, \cdots \,, \, X_n\) is a random sample from a regular statistical model \(\{f_{\theta }(x);\, \theta \in \Omega \}\). A test with critical region \[R = \left \{x:\, S(\theta _0, x) > C\right \}\] where \(C\) is a constant determined by the size of the test, is a locally most powerful test of \(H_0:\, \theta = \theta _0\) against \(H_1:\, \theta > \theta _0\).
Example 4.3.3. Suppose \(X_1, \, \cdots \,, \, X_n\) is a random sample from a \(N(\theta ,1)\) distribution. Show that the locally most powerful test of \(H_0:\, \theta = 0\) against \(H_1:\, \theta > 0\) is also the uniformly most powerful test.
Solution. \(H_0:\, \theta = 0\,\) against \(\, H_1:\, \theta > 0\).
The rejection region for the locally most powerful test is \(R = \{x:\, S(\theta _0,x) > C\}\).
\[L(\theta ) = \prod ^n_{i = 1} \frac {1}{\sqrt {2\pi }}\, e^{-\frac {1}{2}(x_i - \theta )^2} = \left (2\pi \right )^{-n/2}\, e^{-\frac {1}{2}\sum ^n_{i = 1}(x_i - \theta )^2}.\]
\[\mathcal {L}(\theta ) = -\frac {1}{2}\sum ^n_{i = 1} (x_i - \theta )^2 - \frac {n}{2}\, \log 2\pi .\] \begin {align*} S(\theta ) & = (-1)\, \sum ^n_{i = 1}(x_i - \theta )(-) =\sum ^n_{i = 1}x_i - n\theta = n\overline {x} - n\theta = n(\overline {x} - \theta ). \end {align*}
\[S(\theta _0,x)\, > \, C\]
\[n\left (\overline {x} - \theta _0\right )\, > \, C\]
\[n\overline {x}\, > \, C\]
\[\overline {x}\, > \, \frac {C}{n}\]
\[\overline {x}\, > \, K\]
where \(K\) is determined by the size of the test.
Therefore the critical region corresponding to the LMP test is \(R = \{x:\,\overline {x} > K\}\) which is the same as that for the
UMP test. □
Problem 4.3.1. Consider a single observation \(X\) from the LOG\((1,\theta )\) distribution. Find the locally most powerful test of \(H_0:\, \theta = 0\) against \(H_1:\, \theta > 0\).
Show solution
Solution. The logistic density with location \(\theta \) and scale \(1\) is \[f_{\theta }(x) = \frac {e^{-(x-\theta )}}{\left [1+e^{-(x-\theta )}\right ]^{2}} .\] A locally most powerful test of \(H_0:\theta =0\) against \(H_1:\theta >0\) rejects for large values of the score at \(\theta =0\). Now \[\log f_{\theta }(x) = -(x-\theta ) - 2\log \left [1+e^{-(x-\theta )}\right ],\] so \[S(\theta ,x) = \frac {\partial }{\partial \theta }\log f_{\theta }(x) = 1 - \frac {2e^{-(x-\theta )}}{1+e^{-(x-\theta )}} = 1 - \frac {2}{1+e^{\,x-\theta }} ,\] and at \(\theta =0\), \[S(0,x) = 1 - \frac {2}{1+e^{x}} = \frac {e^{x}-1}{e^{x}+1} = \tanh \!\left (\frac {x}{2}\right ).\] This is a strictly increasing function of \(x\), so rejecting for large \(S(0,X)\) is the same as rejecting for large \(X\), and the locally most powerful test is \[R = \left \{x:\ x > c\right \}.\] Under \(H_0\) the standard logistic distribution has \(P_0(X>c) = 1-\dfrac {1}{1+e^{-c}} = \dfrac {1}{1+e^{c}}\), so setting this equal to \(\alpha \) gives \[c = \log \left (\frac {1-\alpha }{\alpha }\right ).\] For \(\alpha =0.05\), \(c=\log 19 \approx 2.944\).
Note. The test is only locally most powerful — best against alternatives just above zero. Here it happens also to be UMP, because the logistic location family has monotone likelihood ratio in \(x\), but that is a bonus rather than something the locally most powerful construction guarantees.
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