A standard technique used by field biologists to estimate animal populations is

Question

A standard technique used by field biologists to estimate animal populations is the capture-recapture method. To be concrete, imagine a lake containing an unknown number N of fish. A sample of size M of these fish is caught, tagged, and released back into the lake. A while later, another sample of size S is taken with replacement, and it is found that X of them are tagged. (a) What is the probability distribution of X? (b) Discuss the assumptions on the nature of fish and the procedure for sampling that you used to justify the distribution above. (For instance, what if tagging a fish is so traumatic that it dies soon thereafter?) (c) What is the Maximum Likelihood Estimator to N? (d) What is the Method of Moments Estimator of N? (e) Are there problems with these estimators? If so, what might you do? (f) What would change in your analysis if the second sample had been taken without replacement?

Explanation / Answer

1.

Total no : of fish = N

X – no : of fish with tag

S- total no : of fish caught

Ratio of fish in sample caught = x / S

Ratio of tagged fish in lake = M / N

We assume ratio is same in sample and lake .

There are so many methods in actual practice based on this kind of assumption

Example is Monte Carlo simulation .

Equating two

We get x = S*M/N .

Now probability distribution that suits above problem is binomial distribution

P = M/N

Q = 1-M/N

B (n,p) = NcM p^M *q^(N-M)               

2 .

For study of fish and other wild animals tagging is a safe method of conducting experiment .

Death of fish will not occur if people employed are experienced and professionals in this field.

3.

Maximum likelihood estimator of binomial distribution is p ;

Check

L = NcM *p^M * q^(N-M) = NcM *p^M * q^(N-M)

Taking log on both sides

Log L = logNcM + Mlogp + (N-M)log q .

Differentiating wrt p on both sides

d(log L ) /dp =   M/p + (N-M)/(1-p)*-1

equating to 0 and rearranging gives

p = M/N

there fore p is MLE of binomial distribution

4.

Method of moments estimator

Mx = E (exp(tx)) = exp(tx)NcM p^M q^(N-M)

= NcM (p+ exp(t ))^M q^(N-M) : in this case x = M

= (q +pexp ) ^ M

MGF about mean

General form of exp(tx) = 1 + tx/1! +(tx)2/2! +(tx)3 /3! + ………………………

Applying it in above equation and rearranging we get

= 1 + Nc1 ( ( t^2/2! *pq + t^3/3! Pq(q-p) + …………………………)

                                                                        + (Nc2 )(t^2/2! *pq + t^2 /3! *pq(q-p ) + ………………….

Now from above function

= mean = Np

= p = M/N .

5 .

If sample size is small then errors will be there with estimators . To avoid that one method is to increase sample size as deviation =z/n , as n increases deviation decreases .

Another method when sample size is low is to use non parametric methods like H test , Spearman Rank Correlation etc .

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