Lecture 13. D) 2-sided Tests and Unbiased Tests
2-sided tests
Consider now the testing problem [math]H_{0}:\theta=\theta_{0}vs.H_{1}:\theta\neq\theta_{0}[/math].
For values [math]\theta_{1}\gt \theta_{0}[/math], we have already seen that the UMP test rejects [math]H_{0}[/math] if [math]\overline{x}\lt k^{'}.[/math]
For values of [math]\theta_{1}\lt \theta_{0}[/math], it is easy to show that the Neyman-Pearson lemma yields a test that rejects [math]H_{0}[/math] if [math]\overline{x}\gt k^{'}.[/math]
Because the suggested UMP test depends on the values of [math]\theta_{1}[/math], the Neyman-Pearson lemma implies that there is a UMP [math]\alpha[/math]-level test does not exist (the lemma is written in the form of “if and only if”, such that if it does not yield a unique test, then a UMP test does not exist).
If one restricts the class of tests further, it is often possible to obtain (a more restricted type of) optimality. We will restrict ourselves to the class of unbiased tests.
Unbiased Tests
An unbiased test satisfies
[math]\text{sup}_{\theta\in\Theta_{0}}\,\beta\left(\theta\right)\leq\text{inf}_{\theta\in\Theta_{1}}\,\beta\left(\theta\right)[/math]
that is, the value of the power function in the null set is always below the value in the alternative set.
Unbiased tests are important because UMP unbiased (UMPU) tests often exist, even when unrestricted UMP tests do not.
A typical example is the case of [math]N\left(\mu,1\right)[/math], with [math]H_{0}:\mu=\mu_{0}vs.H_{1}:\mu\neq\mu_{0}.[/math]
(We will not go into the generalized Neyman-Pearson result applied to the optimal unbiased test.)
In the [math]N\left(\mu,1\right)[/math], it turns out that the UMPU is [math]T\left(X\right)=\left|\overline{X}\right|,[/math] i.e., we reject [math]H_{0}[/math] iff [math]\left|\overline{X}\right|\gt c.[/math]
Consider the following figure, which plots the power functions of the UMP one-sided tests (in orange and blue), and the power function of the two-sided of the UMPU test (in green).
All curves intersect 0.05 at [math]\mu=0[/math], the null hypothesis in this case.
The blue curve is the power function for UMP test [math]H_{0}:\mu=0[/math] vs. [math]H_{1}:\mu\gt 0[/math]: Notice it is relatively very likely to reject [math]H_{0}[/math] when [math]\mu\gt 0[/math].
The orange curve is the power function for UMP test [math]H_{0}:\mu=0[/math] vs. [math]H_{1}:\mu\lt 0[/math]: It is more likely to reject [math]H_{0}[/math] when [math]\mu\lt 0[/math].
Each of these power curves is unsuitable for test problem [math]H_{0}:\mu=0vs.H_{1}:\mu\neq0.[/math]
The green curve is the power function for the two-sided UMPU test. It does extremely well in the regions where the each of the one-sided power functions do the worst. However, notice that it doesn’t do as well where the other power functions do their best. This is expected, since the one-sided tests are UMP.
Intuitively, the two-sided test reallocates rejections by increasing the power function relative to the places where the one-sided tests do worst, at the expense of the cases where they do best.
