R help - Oct 2003 - hypergeometric & population estimates

"help"

We want to estimate the number of caribou in Jasper.  We recently conducted
an aerial survey and saw 70 uncollared caribou and 8 of 11 collared
caribou.  We want to estimate the number of caribou in this population with
95% confidence limits.  Gary White uses the hypergeometric distribution and
determines the population estimates using maximum likelihood and 95%CL as
-2LogLikelihoods.  Below, I determined the population estimate using
dhyper(x,m,n,k) and maximizing the density value as a function of n, but do
not know how I should calculate MLE with this distribution.


x <- 8         # number resighted caribou (white balls drawn)
m <-11         # number collared caribou (white balls total)
k <- 70        #  number caribou seen (# balls drawn)
n <- 1:500     #  ?? unknown number of uncollared caribou (# black balls)

d <- unlist(lapply(n, function(i) dhyper(x,m,i,k)))     # density estimate
for each value of n
data <- data.frame(estimate = n+m, d)
data <- data[is.finite(data$d), ]             # filter out NA's

max.d <- max(data$d)
pop.estimate <- data[data$d == max.d, 1]

Thank-you for your assistance,
Jesse

I'm not sure I understand your notation:
(1) We recently conducted an aerial survey and saw 70 uncollared caribou and
8 of 11 collared caribou.
(2) k <- 70        #  number caribou seen (# balls drawn)

It's the number of balls drawn parenthetical remark that bothers me - I
think the total number of balls drawn should be 78 and the number of
non-white balls drawn is 70.

If
x <- 8         # number resighted caribou (white balls drawn)
m <-11         # number collared caribou (white balls total)
totaldrawn <- 78        #  number caribou seen (total # balls drawn)

I believe that the maximum likelihood estimator you are looking for is given
by

MLE <- floor(m * totaldrawn / x)  #floor(11 * 78 / 8) = 107

I believe the trick is to look at f(n) = P(x|m,totaldrawn,n) as a function
of n and consider the ratio f(n) / f(n-1).  If this ratio is greater than 1
the function is increasing and if the ratio is less than 1 the function is
decreasing.  Then algebraically show that the maximum occurs at floor(m *
totaldrawn / x).

Bytheway, this MLE includes both collared and uncollared balls so it may be
that you are looking for 107 - 11 as your estimate??

hth
Bob
Usual disclaimers...

-----Original Message-----
From: Jesse.Whittington@pc.gc.ca [mailto:Jesse.Whittington@pc.gc.ca] 
Sent: Wednesday, October 01, 2003 12:56 PM
To: R-help@stat.math.ethz.ch
Subject: [R] hypergeometric & population estimates

"help"

We want to estimate the number of caribou in Jasper.  We recently conducted
an aerial survey and saw 70 uncollared caribou and 8 of 11 collared
caribou.  We want to estimate the number of caribou in this population with
95% confidence limits.  Gary White uses the hypergeometric distribution and
determines the population estimates using maximum likelihood and 95%CL as
-2LogLikelihoods.  Below, I determined the population estimate using
dhyper(x,m,n,k) and maximizing the density value as a function of n, but do
not know how I should calculate MLE with this distribution.


x <- 8         # number resighted caribou (white balls drawn)
m <-11         # number collared caribou (white balls total)
k <- 70        #  number caribou seen (# balls drawn)
n <- 1:500     #  ?? unknown number of uncollared caribou (# black balls)
d <- unlist(lapply(n, function(i) dhyper(x,m,i,k)))     # density estimate
for each value of n
data <- data.frame(estimate = n+m, d)
data <- data[is.finite(data$d), ]             # filter out NA's

max.d <- max(data$d)
pop.estimate <- data[data$d == max.d, 1]

Thank-you for your assistance,
Jesse

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