4
$\begingroup$

For the classical simple linear regression model I have derived an hypothesis test for $H_0\colon \left\{\frac{y^*(x^*)-x^*}{\sigma}>1 \right\}$ where $x^*$ is a given value of the covariate $x$ and $y^*(x^*)=a + b x^*$ is the theoretical mean of the response $y$.

I use the test statistic $t=\frac{\hat{y}(x^*)-x^*}{\hat\sigma}$. Something very nice happens: at the boundary $\left\{\frac{y^*(x^*)-x^*}{\sigma}=1 \right\}$ of $H_0$, there is only one possible distribution of the random variable $t$. Hence for a given $\alpha \in ]0,1[$ I can find the critical value $C$ such that the type 1 error of the hypothesis test is exactly $\alpha$ when the rejection rule is $t<C$.

Very nice, but now I want to do the similar hypothesis test for Deming regression, and things are not so nice : the asymptotic distribution of $t$ has several possible distribution at the boundary of $H_0$. Hence I have derived an estimated critical value $\hat C$ and I use the rejection rule $t<\hat{C}$.

Simulations show that the type 1 error is approximately well controlled: for a desired type 1 error $\alpha$ the effective type 1 error is close to $\alpha$. But I wonder whether there are some pitfalls with my procedure ? Do you know other examples where one similarly uses an estimated critical value ? And are there some known pitfalls with these examples ? I think this is not rigorously correct to use a test statistic whose distribution is not uniquely determined at the boundary of $H_0$, but I don't know how to do otherwise.

$\endgroup$
2
$\begingroup$

The original t-table was created using simulation, so there is a clear precident. Most bootstrap and permutation tests use simulation to determine the critical values. Fisher even justified the use of the t-test as a good approximation to the permutation test that did not require as much computation. As long as you are comforatable with the assumptions that go into your simulations then I see no problems with your approach (often I am more comforatable with the assumptions in a simulation (which I know and control) than with the assumptions from a canned procedure (some of which might not be known or apparent)).

$\endgroup$
1
  • $\begingroup$ I'm afraid my question is not clear enough. The problem is not to approximate a critical value using simulations. The problem is that the distribution of $t$ at the boundary of $H_0$ depends on the model parameter, say $\theta$. Hence the critical value $C=C(\theta)$ depends on $\theta$ which is unusual in hypothesis testing, and I use $C(\hat\theta)$ as the critical value. $\endgroup$ Dec 23 '12 at 8:59

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.