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: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] Sent: Wednesday, October 01, 2003 12:56 PM To: [EMAIL PROTECTED] 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 ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help [[alternative HTML version deleted]] ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help
