# Convergence of product of an IID zero and a symmetric random variable

Suppose a super-population model without auxiliary covariates,

$$y_i = \theta + \epsilon_i$$, where $$\epsilon_i \sim N(0,1)$$. Then the score function $$S_i(y_i, \theta) = y_i-\theta$$.

In the case of informative/nonignorable non-response, the probability of non-response suppose $$\pi_i$$ needs to be dependent on $$y_i$$ but I am not considering the exact relation between $$y_i$$ and $$\pi_i$$ like a usually used logistic model $$logit(\pi_i/(1-\pi_i))= \gamma y_i$$ but I am considering $$logit(\pi_i/(1-\pi_i))=\eta_i(y_i)$$ , where $$\eta_i$$ is unspecified function of $$y_i$$ so we do not know the exact relation between $$y_i$$ and $$\pi_i$$ but they are not independent.

Now to estimate the parameter $$\theta$$, one can write sample estimating equations with known response probability $$\pi_i$$ as $$\begin{equation} \frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\pi_i} [S_i(y_i, \theta)] \tag{1} \end{equation}$$

Now using empirically estimated response probability $$\hat \pi_i$$, the sample estimating equation can be written as $$\begin{equation} \frac{1}{N} \sum_{i=1}^{N}\frac{\delta_i}{\hat \pi_i} [S_i(y_i, \theta)], \tag{2} \end{equation}$$

I need to show this estimation equation to be unbiased to derive further properties of estimator $$\hat \theta$$. For unbiasedness I need to show the expectation of (2) directly zero or asymptotically zero.

Since in (2), $$y_i$$ and $$\delta_i$$ are both random variables. After Taylor expansion of $$1/\hat \pi_i$$ around $$\pi_i$$ and first taking expectation on $$\delta_i$$, we have $$\begin{equation} \frac{1}{N} \sum_{i=1}^{N}[S_i(y_i, \theta)] \frac{Var(\hat \pi_i)}{\pi_i}, \tag{3} \end{equation}$$ Now suppose $$G_i=\frac{Var(\hat \pi_i)}{\pi_i}\geq 0$$ and $$S_i=S_i(y_i, \theta)$$ then (3) becomes

$$\begin{equation} \frac{1}{N} \sum_{i=1}^{N}[S_i G_i] \tag{4} \end{equation}$$ Now taking expectation under super population model $$\begin{equation} \frac{1}{N} \sum_{i=1}^{N}E[S_i G_i] \tag{5} \end{equation}$$

As $$S_i$$ is a known function of $$y_i$$ but $$G_i$$ is unspecified function of $$y_i$$ because $$\pi_i$$ is unspecified function of $$y_i$$. Now we can assume $$S_i$$ is IID with $$E(S_i)=0$$ and suppose $$G_i$$ is symmetric. Then I need to show

$$\frac{1}{N} \sum_{i=1}^{N} E(S_i G_i) = 0$$.

Now suppose $$y_i$$ is fixed then both $$S_i$$ and $$G_i$$ are not random variables then I need to show for $$N \to \infty$$,

$$\frac{1}{N} \sum_{i=1}^{N} (S_i G_i) \to 0$$.

Maybe I need to put some conditions on the moments of $$G_i$$ for first question and conditions on both $$S_i$$ and $$G_i$$ for second question.

Any suggestion will be obliged, thanks.