Decompose Covariance by Observations

Question

Suppose I observe $n$ iid realizations of two random variables $X$ and $Y$, denoted respectively $x_i$ and $y_i$. Observations can be groupped into two subsamples, with $n_1$ and $n_2$ observations. I want to decompose the sample covariance $\widehat{\sigma_{XY}}$ by the contribution of each group of observations plus possibly a residual term. Here is what I have:

\begin{eqnarray} \widehat{\sigma_{XY}} & = & \frac{1}{n} \sum^n x_i y_i -n \overline{x} \overline{y} \\ & = & \frac{n_1}{n} \frac{1}{n_1} \sum^{n_1} x_i y_i + \frac{n_2}{n} \frac{1}{n_2} \sum^{n_2} x_i y_i \\ &&- n\left(\frac{n_1}{n} \frac{1}{n_1} \sum^{n_1} x_i + \frac{n_2}{n} \frac{1}{n_2} \sum^{n_2} x_i\right) \left(\frac{n_1}{n} \frac{1}{n_1} \sum^{n_1} y_i + \frac{n_2}{n} \frac{1}{n_2} \sum^{n_2} y_i\right) \\ & =& \frac{n_1}{n} \frac{1}{n_1} \left(\sum^{n_1} x_i y_i -n_1 \overline{x}_1 \overline{y}_1\right) + \frac{n_2}{n} \frac{1}{n_2} \left(\sum^{n_2} x_i y_i -n_2 \overline{x}_2 \overline{y}_2\right) - \frac{n_1n_2}{n}\left(\overline{x}_1\overline{y}_2 + \overline{x}_2 \overline{y}_1\right)\\ & = & \frac{n_1}{n} \widehat{\sigma_{XY, 1}} + \frac{n_2}{n} \widehat{\sigma_{XY, 2}} - \frac{n_1n_2}{n}\left(\overline{x}_1\overline{y}_2 + \overline{x}_2 \overline{y}_1\right) \end{eqnarray}

That is, the overall sample covariance is a weighted average of the sample covariances within each group minus a residual term.

However, in a numerical example this decomposition does not hold as I do not obtain the overall sample covariance on the LHS. Especially the residual term seems too large.

I would appreciate if somebody can double-check whether there is a mistake in the above.

Many thanks

Answer 1

Covariance of $X$ and $Y$ is $\mathrm{Cov}(X,Y) = \mathrm{E}(XY) - \mathrm{E}(X)\mathrm{E}(Y)$, so the estimate $\widehat{\sigma_{XY}} = \sigma$

$\sigma = \frac{1}{n} \left(\sum^n x_i y_i - n \overline{x} \overline{y}\right) = -\overline{x} \overline{y} + \frac{1}{n} \sum^n x_i y_i$

NB: factor $\frac{1}{n}$ does not multiply $\overline{x} \overline{y}$.

It follows that \begin{eqnarray} \sigma & = & \frac{n_1}{n} \sigma_1 + \frac{n_2}{n} \sigma_2 - \frac{n_1 n_2}{n^2}\left(\overline{x}_1\overline{y}_2 + \overline{x}_2 \overline{y}_1 - \overline{x}_1 \overline{y}_1 - \overline{x}_2 \overline{y}_2 \right) \end{eqnarray}