1

Let $X_{1}, X_{2},...,X_{n}$ be i.i.d from the Laplace distribution or Double exponential distribution $DE(\mu, \sigma)$ with the following pdf,

$$f(x) = \frac{1}{2\sigma} e^{\dfrac{-|x-\mu|}{\sigma}}\quad, \quad\mu\in \mathbb{R}, \quad\sigma>0$$

I am intended to find the sufficient and complete statistics for $\mu$ and $\sigma$, It can be done in the following way.

$$f(X_{1}, X_{2},...,X_{n}) = \frac{1}{(2\sigma)^{n}} e^{\dfrac{-\sum^{n}_{i=1}|x_{i}-\mu|}{\sigma}}\quad, \quad\mu\in \mathbb{R}, \quad\sigma>0$$

So, we can say that $(m, \sum^{n}_{i=1}|X_{i}-m|)$ is sufficient and complete statistics for $(\mu, \sigma)$, where $m$ is the median.

I also want to determine the distribution of $T=\sum^{n}_{i=1}|X_{i}-m|$. Do you have any idea to find the distribution of $T$?

Thank you in advance.

score324
  • 325
  • 2
  • 10
  • 2
    You haven't shown that the two statistics in question are complete and sufficient, you've just stated it. – jbowman Jul 11 '18 at 01:34
  • 2
    If $m$ were known $T$ would have a gamma distribution; asymptotically that shouldn't be a bad approximation, but in small samples it will be off. A small adjustment to the shape parameter seems to yield an excellent approximation. By eye it looks like subtracting 1/2+1/(n+2) from the shape works quite well across a broad range of odd sample sizes (the different definition of median for even samples requires another approximation for that case, though that one seems fine for n=6 and up in any case) . This won't be the actual distribution of course. – Glen_b Jul 11 '18 at 02:07
  • You can find an answer by reading this: https://stats.stackexchange.com/questions/163728/a-sufficient-statistic-for-laplace-distribution – kjetil b halvorsen Jul 11 '18 at 10:48
  • @Glen_b, So, I can write $F(T) = P(T\leq t) = P(\sum|X_{i}-\mu|\leq t)$. How to expand this? Do I need to use the iid condition? – score324 Jul 11 '18 at 14:51
  • 1
    There is no non-trivial sufficient statistic for the Laplace distribution since it is not an exponential family. – Xi'an Jul 12 '18 at 21:19
  • And a statistic cannot depend on an unknown parameter, a common mistake found in my 3rd year exams. – Xi'an Jul 19 '18 at 14:18
  • Would it be $Gamma(n-1,\sigma)$? – score324 Jul 20 '18 at 00:46

0 Answers0