[R] MLE with parameters restricted to a defined range using bbmle

[Rolf Turner]

I know nothing about the bbmle package and its mle2() function, but it 
is a general truth that if you need to constrain a parameter to be 
positive in an optimisation procedure a simple and effective approach is 
to reparameterize using exp().

I.e. represent xmin as exp(lxmin) (say) and use lxmin as the argument
to your objective function.

This strategy rarely if ever fails to work.

cheers,

Rolf Turner

On 09/12/14 09:04, Bernardo Santos wrote:
> Dear all,
> I am fitting models to data with mle2 function of the bbmle package.In specific, I want to fit a power-law distribution model, as defined here (http://arxiv.org/pdf/cond-mat/0412004v3.pdf), to data.
> However, one of the parameters - xmin -, must be necessarily greater than zero. What can I do to restrict the possible values of a parameter that are passed to the optimizer?
> Here there is a sample of my code:
> # Loading library
> library(bbmle)
>
> # Creating data
> set.seed(1234)
> data <- rexp(1000, rate = 0.1) # The fit will not be too good, but it is just to test
>
> # Creating the power-law distribution density function
> dpowlaw <- function(x, alfa, xmin, log=FALSE){
>    c <- (alfa-1)*xmin^(alfa-1)
>    if(log) ifelse(x < xmin, 0, log(c*x^(-alfa)))
>    else ifelse(x < xmin, 0, c*x^(-alfa))
> }
> # Testing the function
> integrate(dpowlaw, -Inf, Inf, alfa=2, xmin=1)
> curve(dpowlaw(x, alfa=2.5, xmin=10), from=0, to=100, log="")
> curve(dpowlaw(x, alfa=2.5, xmin=1), from=1, to=100, log="xy")
>
> # Negative log-likelihood function
> LLlevy <- function(mu, xmin){
>    -sum(dpowlaw(data, alfa=mu, xmin=xmin, log=T))
> }
>
> # Fitting model to data
> mlevy <- mle2(LLlevy, start=list(mu=2, xmin=1))
> The result of model fitting here is Coefficients:
>         mu      xmin
> -916.4043  890.4248
> but this does not make sense!xmin must be > 0, and mu must be > 1.What should I do?
> Thanks in advance!Bernardo Niebuhr

-- 
Rolf Turner
Technical Editor ANZJS