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Let the power function be defined as $\beta(\theta)=P_{\theta}(\mathbf{X} \in R)$, where $R$ is the rejection region associated to the test being considered.

Can I state that the Probability of type II error is equal to $\displaystyle 1-\int_{\theta \in \Theta_1} \beta(\theta) \ d\theta$?

Below we see a definition of statistical Power, in Casella and Berger 'Statistical Inference'. enter image description here

I don't seem to be able to understand how defining in this is consistent with the definition of power that I find on Wiki,i.e. Power is the probability of rejecting the null when the alternative is true, unless the answer to my question is 'yes'...

Any help would be appreciated.

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    $\begingroup$ Interesting idea, but no. For example: the integral is not a probability in most cases: we could think of many examples where it is larger than 1. $\endgroup$ Mar 17 '16 at 9:55
  • $\begingroup$ @StijnDeVuyst But isn't the beta function a conditional (on theta) probability? Could you give an example where the integral is greater than 1? Thanks ;) $\endgroup$ Mar 17 '16 at 9:59
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    $\begingroup$ That is exactly the problem: $\beta(\theta)$ is a conditional probability and not a joint probability. From a Bayesian perspective, it would make sense to say $\text{Prob}[X\in R] = \int_{\theta} \text{Prob}[X\in R, \Theta \in (\theta,\theta+d\theta)] $, but $\int_{\theta} \text{Prob}[X\in R | \Theta = \theta] d\theta$ can not reasonably be interpreted as a probability. I'm not saying it is not a useful quantity, but I don't think it is a probability. $\endgroup$ Mar 17 '16 at 10:18
  • $\begingroup$ @StijnDeVuyst I've edited the question to make it more clear my reasoning. Thanks $\endgroup$ Mar 17 '16 at 10:18
  • $\begingroup$ @StijnDeVuyst you're right. but then how can the wiki definition be consistent with the definition in the book? $\endgroup$ Mar 17 '16 at 10:45
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I think we should go back to the idea of hypothesis testing: the parameter $\theta$ has one true value that is unkown. We have some ''idea'' about the value and we want to test whether this ''idea'' about the value is confirmed or rejected by the data.

But the important thing that there is only one true value for $\theta$. All other values are impossible (so probability is zero) , only the one true (but unknown) value is possible. So as @StijnDeVuyst says, there is only one value for $\theta$.

Since you don't know that value, one tries to compute the power for different values for $\theta$ but that is only because we do some kind of ''robustness'' check, we do not know the true value, so we can not compute the true power and therefore we compute it for several values for $\theta$ and analyse the ''power function''.

But know that only one true value exists. So only one value of this function is the 'real' one.

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