I am sampling a population of unknown size and unknown distribution. The sample will be taken over distinct time intervals, but I have to reject any duplicates in the given time interval. The sample sizes over each time interval will vary in size. The sample will then be combined to form the population sample. My goal is to estimate the size of the population.
Example:
Time period 1) $\{x_1,x_4,x_8,x_8,x_8\}$ reject fourth and fifth observation and I am left with $\{x_1,x_4,x_8\}$ as my sample in time period 1.
Time period 2) $\{ x_{10}, x_1, x_7,x_3, x_8, x_{10}, x_{11}, x_7 \}$, reject sixth and eight observations and I am left with $\{ x_{10}, x_1, x_7,x_3, x_8, x_{11} \}$ as my sample in time period 2.
I now combine my sample from time periods 1 and 2 to get $\{x_1, x_2, x_3, x_4, x_7, x_8, x_8, x_{10}, x_{11} \}$
My question concerns the process that I am following. It appears to be selection bias. What can I do to address it? Is there some way to weight the samples with multiple duplicates more heavily when I make my estimation. Are there any other issues I should be aware of?