From previous stages of the question I have calculated the observed values of ln(θ) as

ln(20/23) based on 23 observations

ln(33/40) based on 20 observations

ln(9/11) based on 11 observations

ln(27/32) based on 4 observations

The first estimator is based on a weighted sum of ln(θ) hence;

(ln(θ)_hat) = (23*ln(20/23) + 20*ln(33/40) + 11*ln(9/11) + 4*ln(27/32)) / 58

The second estimator is based on a conventional average hence;

(ln(θ)_hat) = ln(20/23) + ln(33/40) + ln(9/11) + ln(27/32)) /4

Any help on this would be appreciated.