# #StackBounty: #hypothesis-testing #confidence-interval #mean #asymptotics #small-sample What "nice" property of a confidence …

### Bounty: 50

Suppose that I have $$X_{i} overset{i.i.d.}{sim} P$$ with $$E[X_{i}]=mu$$ and $$V[X_{i}^{2}] = sigma^{2}.

Then by the central limit theorem I know that:
begin{align} sqrt{n} (bar{X}{n} – mu) overset{d}{to} N(0,sigma^{2}) end{align}
where $$bar{X}$$
{n}\$ is the sample average. Suppose for some silly reason I know the value of $$sigma^{2}$$. Then this asymptotic approximation allows me to justify confidence sets for $$mu$$ of the form:
begin{align} bar{X}{n} pm q{alpha/2} sqrt{frac{sigma^{2}}{n}} end{align}
where $$q_{alpha/2}$$ is the $$alpha/2^{th}$$ quantile of the standard normal. In particular:
begin{align} lim_{n to infty} P left(q_{alpha/2} leq sqrt{n} frac{(bar{X}{n} – mu)}{sigma} leq -q{alpha/2} right) = 1-alpha\ implies lim_{n to infty} P left(bar{X}{n} + q{alpha/2}frac{sigma}{sqrt{n}} leq mu leq bar{X}{n} -q{alpha/2}frac{sigma}{sqrt{n}} right) = 1-alpha\ end{align}
For simplicity, let:
$$CI_{1} = left[bar{X}{n} + q{alpha/2}frac{sigma}{sqrt{n}} , bar{X}{n} -q{alpha/2}frac{sigma}{sqrt{n}} right]$$
Now suppose that I am a strange statistician, and that rather than the confidence interval constructed above, I prefer a confidence interval (for whatever reason) of my own making:
$$CI_{2} = left[bar{X}{n} + q{alpha/2}frac{sigma}{sqrt{n}}+b_{n} , bar{X}{n} -q{alpha/2}frac{sigma}{sqrt{n}} -b_{n}right]$$
where $$b_{n} = o(n^{-1/2})$$ is some vanishing deterministic sequence. Note that $$CI_{2}$$ also provides $$1-alpha$$ coverage probability asymptotically.

My question: is there any reason to prefer $$CI_{1}$$ to $$CI_{2}$$? Asymptotically they are the same, so I suspect any reason would need to appeal to finite sample arguments. For example, I can always construct the sequence $$b_{n}$$ such that $$CI_{1}$$ and $$CI_{2}$$ are VERY different in finite sample. So what statistical justification would lead someone to use $$CI_{1}$$ versus $$CI_{2}$$? Is there a name for the desirable property $$CI_{1}$$ possesses that $$CI_{2}$$ does not?

Thanks so much!

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