# Convergence of weighted average with fixed variables

I am dealing with administrative data. Suppose I have a finite population of size $N$. The finite population estimating equation that defines the parameter $\theta_0$ can be written as

$$\frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta) \,\,\,\ such \,\, that \,\,\,\,\, \frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta_0)=0 \tag{1}$$

Suppose the score function $S_i(y_i, \theta) = y_i-\theta$ then the parameter of interest $\theta_0=\frac{1}{N} \sum_{i=1}^{N}y_i = \bar Y$. I want to estimate the parameter $\bar Y$ considering nonignorable non-response. Denote the unknown response probability by $\pi_i$ and suppose the empirically estimated non-response probability is $\hat \pi_i$. Then the estimating equation for respondent population is

$$\frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i} S_i(y_i, \theta), \tag{2}$$ where $\delta$ is response indicator and is a random variable, but $y's$ are now fixed like design based approach in survey sampling.

Taking the expectation of (2) over $\delta$, we have $$E_\delta[\frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i}S_i(y_i, \theta)|y]= \frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta)E_\delta[\frac{\delta_i}{\hat \pi_i}] \tag{3}$$ Expanding $1/\hat \pi$ around $\pi_i$ using Taylor expansion and taking expectation over $\delta$, suppose we have $E_\delta[\delta_i/\hat \pi_i]=w_i$ then from (3), we can write

$$E_\delta[\frac{1}{N} \sum_{i=1}^{N}S_i(y_i, \theta)|y]=\frac{1}{N} \sum_{i=1}^{N}w_i S_i(y_i, \theta) = \frac{\sum_{i=1}^{N}w_i}{N} \frac{\sum_{i=1}^{N}w_i S_i(y_i, \theta)}{\sum_{i=1}^{N}w_i} \tag{4}$$ Now $w_i$ are positive values/weights and for $N \to \infty$, $\sum_{i=1}^{N}w_i \to \infty$ and further it can be assumed that for positive constant C, $|w_i|<C$. Then $\frac{\sum_{i=1}^{N}w_i}{N}=O(N)$.

Having $\sum_{i=1}^{N}S_i(y_i, \theta_0)=0$. From (4), first I need to show for $N \to \infty$, the weighted mean
$$\frac{\sum_{i=1}^{N}w_i S_i(y_i, \theta_0)}{\sum_{i=1}^{N}w_i} \to 0 \,\,\, a.s \,\,\, or \,\,\,\, p \,\,\, etc. \tag{5}$$ Then finally for $N \to \infty$, I can obtain $$E_\delta[\frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i}S_i(y_i, \theta)|y] \to 0$$ that I actually I need.

What I have known is that, (5) can be proved for $S_i$ as random variable with $E[S_i(y_i,\theta)]=0$ from page 2 of http://www.ee.bgu.ac.il/~guycohen/clsurvey.pdf

But in my case $S_i$ is fixed variable like explanatory variables in regression. So I cannot take expectation over $S_i$ or $y_i$. I think I need some conditions on $w_i$....

Any suggestion would be highly appreciated.

• By "$X_i$ are not random variables" what do you mean exactly? – Math1000 Oct 21 '16 at 21:40
• Now I hope the question is clear. – ZAH Oct 23 '16 at 20:09