Alternative hypothesis

Why you can't really do it

The difficulty is in creating the sampling distribution for $M\neq E$. In other words, what would be the probability of seeing the data you observed if $M \neq E$. More specifically: what is the probability of seeing your estimate of the parameter if the hypothesis is true.

The usual example is a normal distribution, and an estimate of the mean. In which case, given the mean and variance under the null you can work out how likely estimating various values of the mean are. When working with means, if $\mu_0$ is the mean under the null hypothesis, and $\hat{\mu}$ is the estimation from your data, you can get a probability distribution $D$, so

$$\hat{\mu} \sim D(\mu_0)$$

this is called a sampling distribution. Your $p$-value, or what have you, will be related the probability of your estimate according to this.

If you cannot work out this distribution, then you can't do this kind of test. Basically, one can only do it when the parameters of your hypothetical distribution have exact values, not a range of values. This can only really be done in a few cases, for example a coin toss where $H_1: \text{not heads} = H_1: \text{tails}$.

What you could do instead

There are a couple of solutions:

1. Just go with the $H_0: E=M$ and accept that it is weak thing to do. Remember to report the $\beta$ value (assuming that the null isn't rejected). (usually the standard thing to do)

2. Use a Bayesian approach.

3. Give up on testing completely, and just quantify the difference in an intuitive way. (and use this to supplement 1)

4. Formulate a specific an alternative hypothesis $H_1: E=\text{somthing specific}$, and use a different form of testing - a log likelihood test for example.

5. There's probably something else...