4.3 Locally Most Powerful Tests
A UMP test does not always exist. Alternatives close to the null value are the hardest to detect, so it makes sense to ask for a test whose power rises as fast as possible as \(\theta \) moves away from \(\theta _0\).
Definition 4.3.1. A test of \(H_0\colon \theta = \theta _0\) against \(H_1\colon \theta >\theta _0\) with power function \(\beta (\theta )\) is locally most powerful (LMP) if for every other test of the same size, with power function \(\beta ^*(\theta )\), there is an \(\varepsilon >0\) such that \(\beta (\theta )\ge \beta ^*(\theta )\) for all \(\theta _0<\theta <\theta _0 + \varepsilon \).
Theorem 4.3.2. For a random sample from a regular model, the test with critical region \[R = \{x : S(\theta _0; x)>C\},\] where \(C\) is determined by the size, is locally most powerful for \(H_0\colon \theta = \theta _0\) against \(H_1\colon \theta >\theta _0\).
Sketch of proof. All tests of size \(\alpha \) have \(\beta (\theta _0) = \alpha \), so for \(\theta \) just above \(\theta _0\), \(\beta (\theta )\approx \alpha + (\theta - \theta _0)\beta '(\theta _0)\), and the LMP test is the one that maximises the slope \(\beta '(\theta _0)\). Differentiating under the integral sign, \begin {align*} \beta '(\theta _0) &= \int _R\frac {\partial f_\theta (x)}{\partial \theta }\Big |_{\theta _0}dx\\ &= \int _R S(\theta _0; x)f_{\theta _0}(x)\,dx . \end {align*}
Maximising \(\int _R S(\theta _0; x)f_{\theta _0}(x)\,dx\) subject to \(\int _Rf_{\theta _0}(x)\,dx = \alpha \) is the problem of the Neyman–Pearson lemma with \(f_1\) replaced by \(S(\theta _0; x)f_{\theta _0}(x)\), and the same argument shows that the optimal region is \(\{x : S(\theta _0; x)>C\}\). □
Example 4.3.3. Let \(X_1, \dots , X_n\) be a random sample from \(N(\theta , 1)\). Show that the LMP test of \(H_0\colon \theta = 0\) against \(H_1\colon \theta >0\) is the UMP test.
Solution. \(\mathcal {L}(\theta ) = -\frac 12\sum (x_i - \theta )^2 - \frac n2\ln 2\pi \), so \(S(\theta ) = \sum (x_i - \theta ) = n(\bar x - \theta )\) and \(S(0; x) = n\bar x\). The LMP region \(\{n\bar x>C\}\) is \(\{\bar x>K\}\), the UMP region of example 4.2.3.
Example 4.3.4. Let \(X_1, \dots , X_n\) be a random sample from the Cauchy distribution with location \(\theta \), \(f_\theta (x) = \frac {1}{\pi [1 + (x - \theta )^2]}\). Find the LMP test of \(H_0\colon \theta = 0\) against \(H_1\colon \theta >0\).
Solution. \(\ln f_\theta (x) = -\ln \pi - \ln \left [1 + (x - \theta )^2\right ]\), so \[S(\theta ; x) = \sum _{i=1}^n\frac {2(x_i - \theta )}{1 + (x_i - \theta )^2},\qquad S(0; x) = \sum _{i=1}^n\frac {2x_i}{1 + x_i^2} .\] The LMP test rejects \(H_0\) when \(\sum \frac {2x_i}{1 + x_i^2}>C\). Each term is bounded by \(1\) in absolute value, so a single wild observation cannot force rejection, unlike a test based on \(\bar x\) (which for Cauchy data is no better than a single observation). Under \(H_0\) each term has mean \(0\) and variance \(E\left [\frac {4X^2}{(1 + X^2)^2}\right ] = \frac 12\), so for large \(n\), \(C\approx z_\alpha \sqrt {n/2}\). No UMP test exists here, because the likelihood ratio is not monotone in any single statistic.
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 \(P_0(X>c) = 1-\frac {1}{1+e^{-c}} = \frac {1}{1+e^{c}}\), 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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