Hi guys, I'm stuck on one problem, regarding likelihood functions. I found the MLE of the exponential:
y=(λ)exp[−(λ)x]
And the MLE was found to be:
λ^=1/x
But I'm stuck on answering the question about whether it's unbiased and consistent. I know that for an estimator to be consistent, its variance approaches 0 as n increases, and that the biased formula is:
Bias(X)^=E(X)^−X
So I apply it to my estimator and get this:
Bias[λ]^=E(1/x)−x
So, where do I go from here? My lecturer told me that it can "easily be found by integration" but how? Any help would be appreciated
I've had a think about it, and I don't think E[1/x]=1/E[x] so I think what I've said above is incorrect.
I've had a look on google, and it seems as if you need to use the gamma distribution to find the required bias. I do not think this can easily be found by integration at all?? TBH I think you should speak to your lecturer again because unless I'm missing something, this is a really difficult question!
If you're meant to find the asymptotic bias though, the answer is 0 because MLEs are always asymptotically unbiased.
Sorry, I feel really bad telling you the wrong thing.
I've had a think about it, and I don't think E[1/x]=1/E[x] so I think what I've said above is incorrect.
I've had a look on google, and it seems as if you need to use the gamma distribution to find the required bias. I do not think this can easily be found by integration at all?? TBH I think you should speak to your lecturer again because unless I'm missing something, this is a really difficult question!
If you're meant to find the asymptotic bias though, the answer is 0 because MLEs are always asymptotically unbiased.
Sorry, I feel really bad telling you the wrong thing.