[R] Fitting weibull, exponential and lognormal distributions to left-truncated data.

[Gough Lauren]

Dear All,

I have two questions regarding distribution fitting.

I have several datasets, all left-truncated at x=1, that I am attempting
to fit distributions to (lognormal, weibull and exponential).  I had
been using fitdistr in the MASS package as follows:

fitdistr<-(x,"weibull")

However, this does not take into consideration the truncation at x=1.  I
read another posting in this forum that suggested using the argument
"lower" to truncate the distribution fitting.  However, this does not
seem to be working.  For example, when I attempt to fit a weibull
distribution truncated at x=1 using "lower", it seems to set the
best-fit shape parameter at 1:

> fitdistr(x,"weibull",lower=1)
     shape        scale   
  1.00000000   9.87964337 
 (0.02358731) (0.40649570) ##I have tried this on other datasets also
truncated at x=1 and get the same result (i.e. shape=1).

Does anyone know how to successfully fit the exponential, weibull and
lognormal distributions to truncated data?



Secondly, as my datasets are large (>1000 data points) assessing the fit
of the distribution with kolmogorov smirnov goodness of fit tests is
routinely showing statistical significance for all distributions.
Therefore, I would like to plot the observed data with the theoretical
best fit distributions (weibull, exponential and lognormal) to visually
assess which fits the observed data best.  So far I have been doing this
as follows:

>fitdistr(x,"weibull")
shape        scale   
  a   		b 

>D1<-density(x) ##density distribution of observed data
>D2<-density(rweibull(1500,shape=a,scale=b)) ##density of a random
variable following the theoretical best fit weibull distribution with
shape parameter =a, scale parameter = b.

>plot(range(D1$x),range(D1$y,D2$y),type="n",xlab="x",ylab="Density")
>lines(D1,col="red")
>lines(D2,col="blue")

This successfully plots the two density curves on the same graph, but it
plots data below the x=1 threshold - even for the observed data!  I have
tried limiting the scale of x-axis using xlim=c(1,150) but the graph
still plots the origin of the graph as (0,0).  I can only get different
origins if I limit x more extremely e.g. xlim=c(50,150).  Does anyone
know how I can successfully change the origin of the graph to (1,0)?


Sorry for the long e-mail! Any help would be greatly appreciated.

Regards,

Lauren

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