## Hypothetical Statistics Question

Let $$\phi(x,\mu,\sigma)$$ denote the normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$ evaluated at $$x.$$ Consider the model distribution $$p(x,\mu)=(1-t)\cdot \phi(x,\mu-3t,1)+t\cdot \phi(x,\mu+3\cdot(1-t),t^2),$$ for some known, fixed, but very small $$t>0.$$ Since $$t$$ is small, $$p$$ looks very much like a normal distribution with variance $$1$$ and mean $$\mu,$$ except for a tall spike around $$\mu+3.$$ See the picture below. The mean of $$p$$ is $$\mu.$$ Note in the last term, the variance is $$t^4.$$ The total mass of the spike is $$t,$$ thus small, and so the cumulative distribution for $$p$$ will look very much like the standard cumulative distribution for $$\phi(x,\mu,1).$$

Suppose we have a sample $$x_0,$$ assumed to be from a distribution $$p$$ of the form above, but of unknown mean. To repeat, $$t$$ is known. Consider the hypothesis: $$H_0: \mu=x_0-3\cdot(1-t)$$ Do we reject $$H_0$$?

If $$H_0$$ is true, then $$p(x_0,\mu)\approx t \cdot \phi(x_0,x_0,t^2) = 1/(t \sqrt{2\pi}),$$ and this is very large, about $$1/t$$ times larger than $$p(\mu,\mu).$$ Indeed, $$x_0-3\cdot(1-t)$$ is very close to the maximum likelihood estimate for $$\mu.$$ These seem to be reasons not to reject $$H_0.$$

On the other hand, the overall mass of the spike is small, and the entire spike is well out on the tail of $$p.$$ Thus it is in some sense unlikely to get an $$x_0$$ out there, assuming $$H_0$$ is true. Since the cumulative distribution looks very much like the normal cumulative distribution for $$\phi(x,\mu,1),$$ it makes sense to apply the usual test and reject $$H_0.$$

I find that not rejecting in this case is the right thing to do, but I am not sure what others might think. I also wonder how often such a question is relevant.