R help - Jul 2012 - Maximum Likelihood Estimation Poisson distribution mle {stats4}

Hi everyone!
I am using the mle {stats4} to estimate the parameters of distributions by
MLE method. I have a problem with the examples they provided with the
mle{stats4} html files. Please check the example and my question below!
*Here is the mle html help file * 
http://stat.ethz.ch/R-manual/R-devel/library/stats4/html/mle.html
http://stat.ethz.ch/R-manual/R-devel/library/stats4/html/mle.html 

*In the example provided with the help *
> od <- options(digits = 5)
> x <- 0:10    *#generating Poisson counts*
> y <- c(26, 17, 13, 12, 20, 5, 9, 8, 5, 4, 8)  *#generating the
> frequesncies*
> 
 ## Easy one-dimensional MLE:
> nLL <- function(lambda) -sum(stats::dpois(y, lambda, log=TRUE))  *#they
> define the Poisson negative loglikelihood*
> fit0 <- mle(nLL, start = list(lambda = 5), nobs = NROW(y)) * #they
> estimate the Poisson parameter using mle*
> fit0  *#they call the output*
Call:
mle(minuslogl = nLL, start = list(lambda = 5), nobs = NROW(y))

Coefficients:
lambda 
11.545       * #this is their estimated Lambda Vallue.*

*Now my question is in a Poisson distribution the Maximum Likelihood
estimator of the mean parameter lambda is  the sample mean, so if we
calculate the sample mean of that generated Poisson distribution manually
using R we get  the below!*
> sample.mean<- sum(x*y)/sum(y)
> sample.mean
[1] 3.5433

*This is the contradiction!! *
Here I am getting the estimate as 3.5433(which is reasonable as most of the
values are clustered around 3), but mle code gives the estimate 11.545(which
may not be correct as this is out side the range 0:10)

Why this contradiction? 

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> -----Original Message-----
> > sample.mean<- sum(x*y)/sum(y)
> > sample.mean
> [1] 3.5433
> 
> *This is the contradiction!! *
> Here I am getting the estimate as 3.5433(which is reasonable 
> as most of the values are clustered around 3), but mle code 
> gives the estimate 11.545(which may not be correct as this is 
> out side the range 0:10)
> 
> Why this contradiction? 
> 
> 
Ermm.. the sample mean is mean(y) which is 11.545

I deduce that you and the help page have different views on what the sample y
represents.

i also note that x does not figure at all in the help page's log-likelihood,
suggesting that y is a simple set of counts, whereas you have interpreted x as
the number of instances of each y. That appears not to be the case.

S


 

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On Jul 5, 2012, at 10:48 , chamilka wrote:
> Hi everyone!
> I am using the mle {stats4} to estimate the parameters of distributions by
> MLE method. I have a problem with the examples they provided with the
> mle{stats4} html files. Please check the example and my question below!
> *Here is the mle html help file * 
> http://stat.ethz.ch/R-manual/R-devel/library/stats4/html/mle.html
> http://stat.ethz.ch/R-manual/R-devel/library/stats4/html/mle.html 
> 
> *In the example provided with the help *
>> od <- options(digits = 5)
>> x <- 0:10    *#generating Poisson counts*
>> y <- c(26, 17, 13, 12, 20, 5, 9, 8, 5, 4, 8)  *#generating the
>> frequesncies*
>> 
> ## Easy one-dimensional MLE:
>> nLL <- function(lambda) -sum(stats::dpois(y, lambda, log=TRUE)) 
*#they
>> define the Poisson negative loglikelihood*
>> fit0 <- mle(nLL, start = list(lambda = 5), nobs = NROW(y)) * #they
>> estimate the Poisson parameter using mle*
>> fit0  *#they call the output*
> 
> Call:
> mle(minuslogl = nLL, start = list(lambda = 5), nobs = NROW(y))
> 
> Coefficients:
> lambda 
> 11.545       * #this is their estimated Lambda Vallue.*
> 
> *Now my question is in a Poisson distribution the Maximum Likelihood
> estimator of the mean parameter lambda is  the sample mean, so if we
> calculate the sample mean of that generated Poisson distribution manually
> using R we get  the below!*
> 
>> sample.mean<- sum(x*y)/sum(y)
>> sample.mean
> [1] 3.5433
> 
> *This is the contradiction!! *
> Here I am getting the estimate as 3.5433(which is reasonable as most of the
> values are clustered around 3), but mle code gives the estimate
11.545(which
> may not be correct as this is out side the range 0:10)
> 
> Why this contradiction? 
You misunderstand the setup. y are 11 Poisson counts, x is a dose variable used
further down in the example. So
> mean(y)
[1] 11.54545

Your sample.mean would be relevant if there were 127 count measurements, 26 of
which were 0, 17 were 1, etc. But then the likelihood would be different:
> x <- 0:10
> nLL <- function(lambda) -sum(y*stats::dpois(x, lambda, log=TRUE))
> mle(nLL, start = list(lambda = 5), nobs = NROW(y))
Call:
mle(minuslogl = nLL, start = list(lambda = 5), nobs = NROW(y))

Coefficients:
 lambda 
3.54328 



> 
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> 
> ______________________________________________
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> PLEASE do read the posting guide
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-- 
Peter Dalgaard, Professor
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

Thank you very much Professor .Peter Dalgaard for your kind explanations..
This made my work easy.. I am struggling with this for more than 2 days and
now I got the correct reply. 
Thank again. 

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