Lognormal Distribution

Asked Apr 30 '14 at 00:29

Active Jul 28 '20 at 20:07

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I have a question about MLEs and their regarding to a certain distribution: the lognormal distribution.

$$ f_{X}(x ; \mu, \sigma)=\frac{1}{x \sigma \sqrt{2 \pi}} e^{-\frac{(\ln x-\mu)^{2}}{2 \sigma^{2}}}, \quad x>0 $$ And, I have that the expected value for the distribution is: $$ e^{\mu+\sigma^{2} / 2} $$ The MLE for the expected value and the variance are, respectively: $$ \widehat{\mu}=\frac{\sum_{k} \ln x_{k}}{n}, \quad \hat{\sigma}^{2}=\frac{\sum_{k}\left(\ln x_{k}-\widehat{\mu}\right)^{2}}{n} $$

Now, I'm confused since I need to show if the following are true for my mle for the expected value: consistent

I have that the mle is unbiased, but now, I am not so sure...and since I am dealing with an mle, how am I to show consistency?

Any help or references would be great!

edited Jul 28 '20 at 20:07

UmbQbify

asked Apr 30 '14 at 00:29

H Cruz

1

$\hat{\mu}$ is an estimator of $\mu$ but not of $e^{\mu+\sigma^2/2}$ – Henry Apr 30 '14 at 00:43

Lognormal Distribution

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