(My background: physicist rusty on statistics, evaluating an experimental dataset).

I'm studying a dataset with a random variable $X$, which is approximately normally distributed with standard deviation $\sigma$. I compute the sample average $Y =\sum \frac{X_i}{N}$ and standard error in the mean $\sigma_Y = \frac{\sigma}{\sqrt{N}}$, where $N$ is the number of experiments.

The final step I need to do is to calculate the absolute value $Z = |Y|$ and evaluate the expectation value of $Z$, as well as confidence intervals. I need to consider a general case, since some error bars will be asymmetric ($Y \approx 0$), while others will be symmetric ($Y \gg 0$).

Ideally, I'd like to understand what the confidence intervals should look like, so I don't need to keep sorting the dataset for every data point. Could someone give me some pointers? Thanks a lot!

  • $\begingroup$ What is the mean of $X$? Is it a general unknown $\mu$? Also $Y = X/N$ being a sample average makes no sense. Do you mean $Y = \sum X_i /N$? $\endgroup$ – Greenparker Jun 12 '17 at 8:32

The following method will only work if the mean of $X$ is not 0.

You have have $X \approx N(\mu, \sigma^2)$. Then by the central limit theorem, for the sample average $Y = \sum_{i=1}^{N} X_i/N$, $$ \sqrt{N} (Y - \mu) \overset{d}{\to} N(0, \sigma^2)\,.$$

However, you are interested in $Z = |Y|$. Define a function $g(x) = |x|$. Then $g$ is continuous everywhere and differentiable everywhere except 0. If $\mu \ne 0$, then the Delta Method applies. That is $$\sqrt{N} (g(Y) - g(\mu)) \overset{d}{\to} N(0, \sigma^2 g'(\mu)^2)\, $$ where $$g'(\mu)^2 = \left(\dfrac{dg(x)}{dx} \Big|_{x = \mu} \right)^2 = \left(\dfrac{\mu}{|\mu|} \right)^2 = 1\,.$$

So we have,

$$\sqrt{N} (Z - |\mu|) \overset{d}{\to} N\left(0, \sigma^2 \right)\,.$$

So if $\mu \ne 0$, confidence interval for $|\mu|$ can be created using the above distribution.

  • $\begingroup$ Thanks for the answer, but I'm a bit confused about one aspect. If $\sigma$ is sufficiently large, and $\mu$ sufficiently small, then it seems the confidence interval for $Z$ would include negative numbers, which it shouldn't? $\endgroup$ – malina Jun 12 '17 at 9:58
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    $\begingroup$ @malina Yes it would. It is not ideal, and this is a drawback of large sample confidence intervals for parameters on a restricted space. $\endgroup$ – Greenparker Jun 12 '17 at 10:02
  • $\begingroup$ Do you know if there are other well-defined ways for constructing confidence intervals which ensure the answer is always within the restricted space? $\endgroup$ – malina Jun 13 '17 at 6:55
  • $\begingroup$ @malina Bootstrap confidence intervals most likely. But more than that, I am not so sure. $\endgroup$ – Greenparker Jun 13 '17 at 8:38

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