[R] boostrap astronomy problem

From: Gregory Ruchti <gruchti_at_pha.jhu.edu>
Date: Tue 03 Jan 2006 - 04:29:01 EST


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

gruchti@pha.jhu.edu
Tel: (410)516-8520

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Received on Tue Jan 03 04:34:13 2006

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