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Suppose we have two variables $x$ and $y$ defined in some population, with all values of $x$ known. A Poisson sample is drawn, with corresponding inclusion probabilities $\pi_k$ that are proportional to the values of the variable $x$.Denote the average sample size by $n$ and the realized size of the Poisson sample by $n_p$. Also, denote the total sum of variable by $t_k$.

Find approximate variance of the estimator $$\hat{t} = \frac{t_x}{n_p}\sum_{k \in i}\frac{y_k}{x_k}$$

Solution:

Since inclusion probabilities are proportional to $x$ value, (after some manipulation) we can write $\pi_k = nx_k/t_x$. Then we can rewrite:

$$\hat{t} = n \frac{1}{n_p}\sum_{k\in i} \frac{y_k}{\pi_k}$$

Using Taylor linearization we can approximately write:

$$\hat{t} = t_y + \sum_{k\in i}\frac{y_k}{\pi_k} - \frac{t_y}{n}n_p$$

Now we have two random variables (estimators) and a constant. The first non-constant term is the Horwitz-Thompson estimator of $t_y$ and I know how to get its dispersion. I'm having trouble with the second thing, namely $n_p$. Could somebody help me to estimate its variance and covariance between $n_p$ and H.T. estimator of $t_y$ ? Then my problem would be solved, and hopefully, I would have a better grasp of Poisson sampling.

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