Upper bound for covariance of Hortvitz-Thompson Estimators

Question

I need to bound on a covariance quantity that has come up in a sampling problem. $\widehat{Y}$ and $\widehat{T}$ are Horvitz-Thompson estimators of population totals, $Y=\sum_{i=1}^N y_i$ and $T=\sum_{i=1}^N t_i$, and take the following forms: $$\widehat{Y} = \sum_{i=1}^N \frac{Z_i y_i}{p_i}$$ $$\widehat{T} = \sum_{i=1}^N \frac{Z_i t_i}{p_i}$$

where $Z_i \in \{0, 1\}$ is a binary random variable indicating whether unit $i$ is included in the sample or not, $p_i = P(Z_i = 1)$ is the known probability of unit $i$ inclusion in the sample. $Z_i$'s are not independent and their joint distribution is known. In other words, we know the quantities $p_{ij} = P(Z_i=1 \,\&\, Z_j=1)$. Finally, we can assume $y_i$'s and $t_i$'s are finite and positive.

Can one find a minimal $A$ that satisfies the expression below? $$ Var(\widehat{T})Var(\widehat{Y}) - 2E[\widehat{T}] Cov(\widehat{Y}, \widehat{Y}\widehat{T}) + A \ge 0 $$ The quantity above shows up in expectation of a variance estimator and satisfying the inequality above implies the estimate is conservative (in expectation). Ideally I need $A$ to be in terms of simple quantities for which I have unbiased or conservative estimates (e.g. I have an unbiased estimate for $E[\widehat{Y}^2]$ or $E[\widehat{T}]$ but I don't have a good estimate for $Cov(\widehat{Y}, \widehat{T})E[\widehat{Y}]$).

This problem is equivalent to finding an upper-bound on the following covariance quantity:

$$Cov(\widehat{Y}, \widehat{Y}\widehat{T})$$

I tried to use Young inequality $ab \le \frac{a^2}{2} + \frac{b^2}{2}$ and Cauchy-Swartz, but both ended up with a complicated expression involving 3rd or 4th moments. Appreciate any insights.

Answer 1

As pointed out in the comments, in general the covariance could be infinite, so the upper bound might be infinity. Nevertheless, it might benefit you to consider the following decomposition (which assumes that all relevant quantities are finite):

$$\begin{align} \mathbb{C}(Y,YT) &= \mathbb{E}(Y^2 T) - \mathbb{E}(Y) \mathbb{E}(YT) \\[6pt] &= \mathbb{E}(Y^2 T) - \mathbb{E}(Y) [ \mathbb{E}(Y) \mathbb{E}(T) + \mathbb{C}(Y,T) ] \\[6pt] &= \mathbb{E}(Y^2 T) - \mathbb{E}(Y)^2 \mathbb{E}(T) - \mathbb{C}(Y,T) \mathbb{E}(Y) \\[6pt] &= \mathbb{E}(Y^2 T) - \mathbb{E}(Y^2) \mathbb{E}(T) + [\mathbb{E}(Y^2) - \mathbb{E}(Y)^2] \mathbb{E}(T) - \mathbb{C}(Y,T) \mathbb{E}(Y) \\[6pt] &= \mathbb{C}(Y^2, T) + \mathbb{V}(Y) \mathbb{E}(T) - \mathbb{C}(Y,T) \mathbb{E}(Y) \\[6pt] &= \mathbb{C}(Y^2, T) - \mathbb{C}(Y,T) \mathbb{E}(Y) + \mathbb{V}(Y) \mathbb{E}(T). \\[6pt] \end{align}$$

This decomposition allows you to express $\mathbb{C}(Y,YT)$ in terms of the underlying moments of $Y$, $Y^2$ and $T$. It also "separates" $Y$ and $T$ in the sense that you no longer have a term $YT$ (though you do still look at covariances between these parts). If you are willing to specify relevant moment parameters for the $Y_i$ and $T_i$ values (and specify if they are independent) then you might be able to obtain the relevant moments in the decomposition, which would let you use this decomposition to compute the covariance at issue.