Hi,

I am an astronomer and somewhat new to boostrap statistics. I understand

the basic idea of bootstrap resampling, but am uncertain if it would be

useful in my case or not. My problem consists of maximizing a likelihood

function based on the velocities of a number of stars. My assumed

distribution of velocities of these stars is:

---------------------------------------------------------------------------

dmy=function(x,v,k,t)(k+1)/(v-t)^(k+1)*(v-x)^k

---------------------------------------------------------------------------

where x would be my stellar velocities. (Essentially it is a beta

distribution.)

---------------------------------------------------------------------------

My likelihood function looks something like this:

lm<-function(x){

e<-x[1]

k<-x[2]

log(e) - n*log(k+1) + (k+1)*n*log(e-t)-k*sum(log(e-vg))

}

--------------------------------------------------------------------------

The quantities n and t are known, and vg is my velocity data. I am

minimizing this function using the function "optim" (BFGS option) to find

k and e that minimize this. Also, my data set is small, only about 50

stars. Therefore, I was thinking that I could use the boot function to

resample my data and solve the minimization for each resample. This way I

believe I'll get better estimates for standard errors and confidence

intervals. Is it safe to assume that the distributions for k and e are

approximately Normal, therefore making the bootstrap useful? I have

actually used the boot function with this set up:

--------------------------------------------------------------------------

mystat=function(s,b){

#Negative Log Likelihood Function

lm<-function(x){

e<-x[1]

k<-x[2]

log(e) - n*log(k+1) + (k+1)*n*log(e-t) -

k*sum(log(e-s[b]))

}

#Gradient of Negative Likelihood Function

glm=function(x){

e<-x[1]

k<-x[2]

c(1/e + (k+1)*(n/(e-t)) - k*sum(1/(e-s[b])),-n/(k+1) +

n*log(e-t) - sum(log(e-s[b])))

}

optim(c(480.,2.),lm,glm,method="BFGS",control=list(maxit=10000000))$par

}

#Compute Bootstrap replicates of escape velocity and kr

m2B2=boot(vg,mystat,5000)

------------------------------------------------------------------------

Does this appear to be correct for what I'd like to achieve? I have

looked at the distribution and it appears to be about Normal, but can I

say that this is true for the sampling distribution as well? Also, the

bootstrap distribution is fairly biased, should I be using "bca" or tilted

bootstrap confidence intervals? If so, I am having some trouble getting

the tilted bootstrap to work. Specifically, it is having trouble finding

"multipliers".

Also, should I be in some way taking into account my velocity distribution

when resampling? Any suggestions would be very helpful, thanks.

Thank you for your time.

Greg Ruchti

--

Gregory Ruchti

Bloomberg Center for Physics and Astronomy

Johns Hopkins University

3400 N. Charles St.

Baltimore, MD 21218-1216

[hidden email]
Tel: (410)516-8520

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