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In statistics, how can you tell whether an estimator is robust or not? I need to discuss whether the maximum likelihood estimators (MLE) of the normal distribution are robust or not. The MLE are

$$\hat{\mu} = \begin{pmatrix} \sum_{i=1}^n x_i/n \\ \sum_{i=1}^n y_i/n \end{pmatrix}$$

and for the variances

$$\hat{\sigma^2}=\begin{pmatrix} \sum_{i=1}^n (x_i-\mu_X)^2/n \\ \sum_{i=1}^n (y_i-\mu_Y)^2/n \end{pmatrix}$$

Do this has anything to do with that the estimator is unbiased/biased? Thanks in advance.

Rodrigo de Azevedo
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aa_x
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2 Answers2

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Robustness is sort of a subjective matter. In a nutshell, if you produce an estimate with a robust estimator, and then you add a very extreme data point and re-estimate, you shouldn't produce an estimate that is too different from your first estimate.

What does "extreme" mean? What does "too different" mean? This is precisely where the ambiguity comes in.

One popular way to measure extremity in a regression, for example, is through calculating the influence of each point (look this up). Another way is by checking the jacknife residauls or the cook's distances of the points. All of these methods were developed heuristically. In most cases, statisticians use them all simultaneously to make statements about the robustness of the estimators.

Once you determine how you want to define extremity, judging if the estimator produces a result that is "too different" boils down to a hypothesis test. Some of these are classical parametric tests (i.e. if you know the distribution of the index you used to measure extremity) and others are empirical (like a parametric bootstrap).

In general, bias does not play a role in determining how an estimator will react to extreme data, but consistency might (think about this for the case where you have large N).

For the MLE, what mathematical tools did you use to produce the estimate? Are the estimates sensitive? These are the questions you should be asking.

Mustafa Eisa
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I recommend first reading the Wikipedia article for robust statistics.

After doing so, what do you intuitively conclude about the estimators you described above? Would you say they are robust or not? Could you furnish an example of an estimator that would be more robust than the MLEs?

heropup
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  • I will say that the mles of the means are not robust, because if we have an outlier, then the mle of the means will behave after this. (I don't know if this is making sense for you). I am not sure about the mles of the variances though. – aa_x Jan 12 '15 at 18:38