[R] Standard Error

[Ruben Roa]

> -----Original Message-----
> From:	r-help-bounces at stat.math.ethz.ch [SMTP:r-help-bounces at stat.math.ethz.ch] On Behalf Of Mark Miller
> Sent:	Thursday, November 17, 2005 10:16 AM
> To:	r-help at stat.math.ethz.ch
> Subject:	[R] Standard Error
> 
> I have worked out that when I fit data I get an estimate and a standard error, 
> but all the definitions I can find describe the standard error of a sample as 
> the standard deviation over the square root of the sample size, so if I am 
> fitting to a log-normal distribution, what is the standard error associated 
> with the standard deviation and why is it different from the standard error 
> of the mean.
------------
One thing is the standard error of the 
estimate_of_a_mean_ 
         from 
a random_sample_from_a_population,
whose formula you mentioned. 
Another thing, though related of course, is the standard error of a 
parameter_estimate 
         from 
a model.

The standard error of a parameter estimate from a model is a measure 
of the precision with which the parameter was estimated. The standard 
lognormal distribution is a model with two parameters (there is another 
with three parameters): the mean and the standard deviation. When you 
fit that model -the lognormal distribution- to a sample, you are estimating 
these two parameters. If you maximise the likelihood for your data as a 
function of the two parameters the estimation process, if successful, will 
produce the two estimates and the corresponding standard errors of those 
estimates (plus the estimated covariance between the estimates). Both 
parameters, the lognormal mean and the lognormal standard deviation, are 
unknown and are estimated so that each one has its corresponding measure 
of precision. 

You can think of the standard error of a parameter estimate from a model at 
least in two ways.
(1) Because maximum likelihood estimates tend to distribute normally, then 
the standard errors of parameter estimates are the standard deviation parameter 
estimates in a normal distribution whose mean is estimated by the maximum 
likelihood estimate itself. For example the output report from the ADMB statistical
system simply put the header Standard Deviation in the column for standard errors
of parameter estimates. Presumably this is because the ADMB's author subscribe
to this interpretation.
(2) You can also think of standard error of parameter estimates as measuring the 
curvature of the likelihood function about the maximum likelihood estimate.
In the pure-likelihood theory of inference this is the preferred interpretation. So 
A.W.F. Edwards (1972, Likelihood, Cambridge UP) has renamed the standard 
errors calling them "the span".

I hope this makes sense to you.

Ruben