Median and MVUE

Attention Conservation Notice: optimality in a one-dimensional parametric model is so 1960s. Wikipedia says “For practical statistics problems, it is important to determine the MVUE if one exists, since less-than-optimal procedures would naturally be avoided, other things being equal”, but they’re not using the word “practical” to mean what I would use it to mean

In the Laplace (or double exponential) distribution, with density \[f(x;\theta) =\frac{1}{2} \exp(-|x-\theta|)\] the maximum likelihood estimator of \(\theta\) is the median. Clearly the median is Unbiased. Is it the UMVUE? The other U is easy: if it’s MVUE for any \(\theta\) then it’s obviously MVUE for every \(\theta\) just by symmetry. Is it minimum variance, though? There’s surprisingly little on the Google; you might have thought this would be a popular example or counterexample in mathematical statistics, but it isn’t.

One of the reasons it’s an interesting example is that the median is not a sufficient statistic. We know it’s not a sufficient statistic because among distributions where the range doesn’t depend on the parameter value¹ the only ones with finite-dimensional sufficient statistics are the exponential family models².

We also know that the median is not fully efficient; it does not achieve the Cramér-Rao bound in finite samples.³ This paper gives an unpleasant formula for the exact variance, and shows that it’s noticeably larger than the Cramér-Rao bound even in quite large samples. By itself this doesn’t show that the median isn’t MVUE, since we don’t necessarily expect the bound to be attainable.

The same paper also claims that the mean of the middle three observations can be better than the median. That’s nice and easy to test by simulation! I tried it with \(n=15\)

suppressMessages(library(VGAM))
rr<-replicate(1000000, {x<-sort(rlaplace(15)); 
    c(median=x[8],threedian=sum(x[7:9])/3, mean=sum(x)/15)})

t.test(rr[1,]^2)

## 
## 	One Sample t-test
## 
## data:  rr[1, ]^2
## t = 584.37, df = 1e+06, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  0.09625834 0.09690622
## sample estimates:
##  mean of x 
## 0.09658228

t.test(rr[2,]^2)

## 
## 	One Sample t-test
## 
## data:  rr[2, ]^2
## t = 601.15, df = 1e+06, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  0.08962650 0.09021284
## sample estimates:
##  mean of x 
## 0.08991967

t.test(rr[3,]^2)

## 
## 	One Sample t-test
## 
## data:  rr[3, ]^2
## t = 674.91, df = 1e+06, p-value < 2.2e-16
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  0.1331128 0.1338882
## sample estimates:
## mean of x 
## 0.1335005

We can see that the median and threedian are better than the mean, and the threedian is very slightly better than the median. So the median is not the UMVUE.