# Linearization of an estimator?

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.