# Mean has lower standard error than 5% trimmed mean?

I'm investigating using a trimmed mean to measure the location of various distributions. The distributions sometimes are heavily contaminated and sometimes not. Usually they follow something similar to a log-normal or possibly mixed log-normal distribution, but often the data is "all over the place".

I've looked at the mean, 5% trimmed mean, 10% trimmed mean and 20% trimmed mean. For each I estimate the standard error using the bootstrap.

What I've found surprising though is that according to the bootstrap the mean often has a lower standard error than the 5% trimmed mean. So across a large number of datasets I have found that from the lowest standard error to the highest is 20% trimmed, 10% trimmed, mean, 5% trimmed.

Is this result atypical, or is it something that is commonly seen ? (Note that I am a beginner with respect to robust statistics and the bootstrap, so it is possible I'm making a fundamental conceptual mistake). Thanks for any hints.

Followup results: I reran the exercise but with much more data. In total there were around 4000 datasets I applied the bootstrap to. The results were as follows

technique      number of times lowest std error
mean           1867
5% trimmed     263
10% trimmed    430
20% trimmed    787
median         663


In this new data when the mean has the lowest standard error it is only better by a small amount, whereas when it does badly it performs really poorly. So when I look at the average standard error across all datasets for the different techniques the results are perhaps in line with what would be expected.

technique      avg std error
mean           4.51
5% trimmed     4.33
10% trimmed    4.05
20% trimmed    3.78
median         4.36

• Can I check - when you do the bootstrap, do you trim first, then bootstrap the trimmed sample, or are you bootstrapping the trimming as well as the mean? It is often/sometimes better not to resample the trimming, but to only resample to trimmed data. (This is because the trimming is based on a quantile estimate, and bootstrap can have problems with quantile estimates) – Corone Sep 9 '13 at 12:17
• That sounds like changing the question because you didn't like the answer.... – Nick Cox Sep 9 '13 at 13:54
• @Corone I resample the original data and then trim. – Antonio2100 Sep 15 '13 at 22:57
• @NickCox haha, very good point. Should make it clear, that you should decide your procedure BEFORE you look at the results! – Corone Sep 16 '13 at 9:24

If the underlying population is normally distributed without contamination then the sample mean is the best unbiased estimate (in the sense of the lowest mean square error) of the centre of the population distribution.

This is not always the case with other distributions, which might include those with contamination. So your observation depends on the particular distribution and contamination.

This does seem surprising at first sight, but here is a guess at what is happening.

Focus on what a bootstrap sample is, namely a sample with replacement. So, every now and again some of these samples will include repetitions of the outliers or wild values. Those samples will be trimmed, but in some cases the trimming will not be enough to exclude all the repeated wild values. But as the degree of trimming increases, this pathology is less likely to be seen.

To spell it out, let's imagine a sample of 20 values 1(1)19, 2000. Trimming 5% is always enough to deal with the outlier in the original data. But trimming 5% won't be enough to deal with bootstrap samples with 2000, 2000 or 2000, 2000, 2000 and so on. There will be plenty of cases with no occurrences of 2000, but they (evidently) don't balance the others.

Bootstrapping is of course not white magic that works regardless. With enigmatic output you need to look beyond printed summaries and see what the entire distribution looks like from all your bootstrap samples. My guess is that you have a tail of really wild results at 5% and this is widening the standard errors. In fact you will have tails of really wild results at all trimming proportions, but less marked as the trimming proportion increases.

Otherwise put, part of the problem is that standard error inevitably is influenced by all values, here all trimmed means. I'd look at percentile-based confidence intervals too.

The efficiency of the trimmed mean depends on the shape of the distribution.

If the underlying distribution is very asymmetric -- say, Exponential -- then trimming will bias your mean in the negative direction.

Or, say, if the distribution is a mixture of two distributions with different means, trimming could remove more of one, again biasing the estimate. For instance, if 90% of your data are $N(0,1)$ and the remainder are $N(1,10)$, then trimming will remove most of the latter points, getting you an estimate closer to $0$ than to the true value $0.1$.

So, it's reasonable that the mean should do better than the trimmed mean, even outside of the standard, Normal case.

What seems more suprising is that the accuracy is not monotonic in the amount of trimming -- you list 20%, 10%, 0%, 5% from most to least accurate. This might happen if, say, you had a mixture again, this time 85% of $N(0,1)$ and 15% of $N(0,20)$, since trimming the 5% tails would greatly reduce the sample size of your $N(0,20)$ samples, leading to a high standard error, but trimming enough more would remove them entirely; since they have the same mean you get a better estimate.