The MVUE estimator of the population mean of the raw data is a function
of both the sample mean of the logs and the sample variance of the
logs.
The classic paper is:
Finney, D. J., 1941, "On the distribution of a variate whose logarithm
is normally distributed". J. R. Stat. Soc. Suppl., 7, 155-161
In a regression context see:
Bradu, D. & Y. Mundlak, 1970."Estimation in lognormal linear models",
JASA, 65(320), 198-211
For applications in environmental sciences:
Gilroy, E. J., et al, 1990, "Mean square error of regression-based
constituent transport estimates", WRR, 26(9), 2069-2077
(Other references can be found in this paper also.)
We discuss these and other interesting applications in our short course,
details of which can be found at:
http://www.practicalstats.com/
tra wrote:
>
> Jean-Michel,
>
> I have been thinking about your question for some time.
>
> If the data are log-normally distributed, and you wish to give some summary
> statistics for the distribution, I would usually prefer to quote the median
> and quartiles to give a summary of the shape of the distribution.
> If the analysis e.g. comparison of treatments was based on comparing the
> means of the log-transformed values then I will quote the geometric mean
> rather than the median.
>
> There are circumstances, however, where you need to estimate the arithmetic
> mean. In this case, what is the best estimator of the mean of a log-normal
> distribution if you have a sample of n values?
> Suppose y ~ N(mu, sigma**2) and z = exp(y).
>
> I do not know what is the 'best' but two obvious estimators are:
>
> (i) the arithmetic mean of the sample: Sum(z)/n
> and
> (ii) the maximum likelihood estimate exp(muhat + 0.5*sigmahat**2)
> where muhat and sigmahat are the maximum likelihood estimates of the mean and
> std of the log-transformed values.
> muhat = Sum(y)/n
> sigmahat**2 = Sum((y-muhat)**2)/n
>
> Which of these is best?
>
> I have found a formula for the mean-square-error of these two estimators.
> The ml estimator has smaller MSE for sufficiently large n
> The MSE of the ll estimator is infinite if n < 2*sigma**2
>
> The ml estimator is slightly biassed for finite n.
> If the usual 'unbiassed' estimate of sigma**2 (ie Sum((y-muhat)**2)/(n-1) is
> used then the bias is worse.
>
> This suggests some more theoretical questions that the allstatters may be
> able to shed some light on:
> (i) Is there a 'best' estimator for the arithmetic mean
> (ii) Are maximum likelihood estimators always atleast as good as other
> estimators for suffciently large n
> (iii) How large does n have to be?
>
> Best wishes
>
> Tim Auton
>
> --
> T R Auton PhD MSc C.Math
> Head of Biomedical Statistics
> Proteus Molecular Design Ltd
> Beechfield House
> Lyme Green Business Park
> Macclesfield
> Cheshire SK11 0JL
> UK
> email: [log in to unmask]
>
> Jean-Michel Lemieux wrote:
>
> > Hello,
> > I have a log normal distributin and i would like to know which is
> > the best estimator of the mean. Is it the arithmetic mean (i don't think
> > so), the median or the mean when the datas are transform in log?
> >
> > Thank you very much
> >
> > Jean-Michel
> > __________________________________________________________________
> > Jean-Michel Lemieux
> > [log in to unmask]
> > * * * * * *
> > Département de géologie et génie géologique
> > Universite Laval
> > Québec, Canada
> > G1K 7P4
--
Edward J. Gilroy,PhD [log in to unmask]
13453 W. Oregon Ct.
Lakewood, Co, 80228 phone:303-986-4944
http://www.practicalstats.com/
Applied Statistical Knowledge for
Environmental Judgement & Guidance
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