Derived N and variance from Huggins models with covariates

[Urge4Sturg]

I am seeking help to understand how to derive N.hat and its variance at specific covariate values with a Huggins model.

I have a model with body size as an individual covariate, and I would like to calculate the derived estimates and variances of N.hat for specific sizes.

As I understand it, the basic estimator for N in the Huggins framework is:

Mt+1(tot. number captured) / p* (prob. an individual is captured at least once) [1-(1-p1)*(1-p2)...(1-pt)]

To determine the N.hat for a specific covariate value, I find the real parameter estimates (p's) for that covariate value, calculate p* for that covariate value and divide the total number of individuals captured with that covariate value by p*.

When I derive N.hats for each 1-cm size class and sum over all the sizes observed, I get pretty close to what MARK reports for the derived N.hat (within 4 individuals in a estimated population of 268,000), but I notice the derived N.hat reported by MARK has numbers to the right of the decimal point. How is MARK calculating the N.hat differently?

For variance estimates, I used the delta method to first obtain a variance for p* (with the vcv matrix for the real parameters, evaluated at each covariate value) and then use the delta method again to obtain a variance for N.hat with the variance of p*.

When I do this, the se and confidence limits I get for the total N.hat is farther off (approximately 8,000 individuals narrower on either CL). I also tried the variance estimator from Huggins 1989 [ Mt+1 x p*^-2 x (1-p*) ], but that gives me a smaller variance than either MARK gives or the delta method. Should I be using the asymptotic variance estimator?

Am I correct in thinking that the ad-hoc delta method is only an approximation of the N.hats and variances, and that I am overlooking some serious calculus? I have read Huggins 1989 several times, but I must admit I don't follow math notation very well, nor remember much calculus.

Any help to improve my understanding would be greatly appreciated!