# Null Hypothesis with a Strict Inequality

On page 300 in Roussas' textbook "An Introduction to Probability and Statistical Inference" he states the null hypothesis for a test as $H_0: \theta>0.0625$ and the alternate hypothesis as $H_A: \theta \leq 0.0625$. What value of $\theta$ does one use to calculate the distribution of the test statistic under the null hypothesis in this test?

• The issue here is not that there is a "strict inequality", but that the null hypothesis is composite in the sense that it comprises more than one possible distribution of a minimal sufficient statistic. The standard theory of this situation is that of uniformly most powerful (UMP) tests. – whuber Mar 6 '15 at 15:11
• If H0 had been expressed as theta > or = to 0.0625 I would use theta = 0.0625 to calculate the power function. Since there is no equality, I can't do that. The question is what value of theta do I use to calculate the power function for this null hypothesis? – Thomas Mar 6 '15 at 15:21
• The distinction is of no consequence. When the space of possible distributions is partitioned into two sets $\Theta_0$ and $\Theta_1$ corresponding to the null and alternative hypotheses, respectively, the principle you are following is to use the worst value of $\theta$ in $\Theta_0$. This worst value--which is a supremum rather than a maximum--might not be attained by any particular element of $\Theta_0$. Under normal regularity assumptions, it will be attained within the closure of $\Theta_0$. The closure of $\{\theta\,|\,\theta\gt 1/16\}$ is the set $\{\theta\,|\,\theta\ge 1/16\}$. – whuber Mar 6 '15 at 15:27
• Thank you for the explanation. I asked this question because I have been taught to always put the equality in the null hypothesis and this is the first time I have seen things done otherwise. You seem to be telling me that it is technically OK to use theta = 0.0625 in the calculation anyway even though the hypotheses are not stated that way. I will take that to be the short answer to my question. – Thomas Mar 6 '15 at 15:45
• I think your interpretation is good. The answer posted by @behzad.nouri is more accurate, though: the endpoint $\theta=1/16$ is the one to use only when the null hypothesis has a particular form. (This motivates the interest in monotone likelihood ratio families mentioned in the Wikipedia article on UMP tests.) In more general situations it is not always the case that the endpoint is the one to use. – whuber Mar 6 '15 at 15:50

Value of $\theta$ which makes it least likely for the null-hypothesis to be rejected; i.e.

$$\underset{.0625\,<\,\theta } {\mathrm{argmax}} \,\, L(X; \theta)$$

where $L(X; \theta)$ is the likelihood of observation $X$ with parameter $\theta$. Note that for the purposes of hypothesis test, in fact you do not need to solve above for $\theta$ but all you need is

$$\underset{.0625\,<\,\theta} {\mathrm{max}} \,\, \mathrm{P}(X;\theta)$$

which may exists in terms of limit even though $.0625\,<\,\theta$ is not a closed set.

By doing so, you can guarantee that if you manage to reject the null-hypothesis with $1 - \alpha$ significance, then you have at least $1 - \alpha$ confidence that the null hypothesis is not true, because any other value of $\theta$ under the null will give even more confidence in rejecting the null hypothesis.

• can you suggest a specific statistics textbook that discusses what you have presented here? Thanks. – Thomas Mar 9 '15 at 8:48