Suppose I have an MLE estimate $\hat{\theta}$ for a parameter $\theta$, and $\hat{\sigma}$ is the sqrt of the inverse of the negative of the Hessian of the log likelihood at $\hat{\theta}$. Can I write $\hat{\theta} \pm \hat{\sigma}$?

In general, if I have an estimate $\hat{\theta}$ for a parameter $\theta$, and an estimate $\hat{\sigma}$ of the standard deviation of $\hat{\theta}$, can I write $\hat{\theta} \pm \hat{\sigma}$? Will others understand what I mean? Is it also what others write?

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    $\begingroup$ under certain regularity conditions MLEs are asymptotically normal. $\endgroup$ – bdeonovic Apr 18 '14 at 2:20

It's fairly common, but not standard enough that it needs no explanation. $\hat\theta\pm 1.96 \hat\sigma$ is also common—an (approximate perhaps) 95% confidence interval.


I think it depends what you want to convey.

To report the standard error or your estimate you could just write $\hat{\theta} (\hat{\sigma})$. This notation is quite common in the statistical literature, but for the sake of clarity you might mention that the standard error is in parenthesis/brackets the first time you use this notation.

To provide a confidence interval with your estimate you might want to write $\hat{\theta} \pm z_{\alpha/2}\hat{\sigma}$, where $z_{\alpha/2}$ is the quantile of the standard normal distribution and $1-\alpha$ the confidence level of your choice. A common choice is $\alpha = 0.05$ leading to $\hat{\theta} \pm 1.96\hat{\sigma}$.

The second option is a much stronger statement. In addition of providing the standard error of the estimate, it gives a confidence interval relying on the normality of the MLE, which might be too coarse in finite samples: it might include invalid parameter values (e.g., probabilities larger than one), and/or wrongly imply that the confidence interval is symmetric.

  • $\begingroup$ That's a really good point: the nomenclature (symbol-clature?) should be appropriate for the audience. I have never seen the subscript-sigma notation in psychology or biology paper, even it is apparently used frequently in other fields. $\endgroup$ – Matt Krause Apr 23 '14 at 15:15
  • $\begingroup$ @Matt Krause Indeed, the notation might be field-specific. Please note that in the notation I mentioned the standard error is not a subscript, but rather in parenthesis (e.g., $4.2(0.2)$). $\endgroup$ – QuantIbex Apr 23 '14 at 16:16
  • $\begingroup$ I've seen that in tables, but I think I would be baffled if I just stumbled across "The average temperature was 4.2(0.2)C" or whatever in a paragraph. All the more reason to explain your notation! $\endgroup$ – Matt Krause Apr 24 '14 at 18:17

There are lots of parameters that are written as $x \pm y$. Off the top of my head, I have seen people use this format to report:

  • mean and standard error:
  • mean and standard deviation
  • mean and confidence interval
  • mean and range/tolerance (particularly on blueprints or design documents)
  • median and mean absolute deviation.

All of these are fine, as long the values are clearly described Some version of the APA style require explicit labels like "The line's slope was close to unity ($M\pm SE: 1.01 \pm 0.05$). That can be handy, but it can clutter up the text. If you're reporting a lot of values, a simple note ("All data is shown as mean $\pm$ standard errors") can work too.

Obviously, also consider the point you want to make. Standard deviation/MAD/ranges hints at the shape of a distribution, but not about the precision of an estimate. Standard error tells you how precise the estimate is, but does not describe the shape of the distribution.


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