Weighted mean

Variance of the weighted mean (π-estimator for ratio-mean)

The previous section dealt with estimating the population mean as a ratio of an estimated population total (${\displaystyle {\hat {Y}}}$) with a known population size (${\displaystyle N}$), and the variance was estimated in that context. Another common case is that the population size itself (${\displaystyle N}$) is unknown and is estimated using the sample (i.e.: ${\displaystyle {\hat {N}}}$). The estimation of ${\displaystyle N}$ can be described as the sum of weights. So when ${\displaystyle w_{i}={\frac {1}{\pi _{i}}}}$ we get ${\displaystyle {\hat {N}}=\sum _{i=1}^{n}w_{i}I_{i}=\sum _{i=1}^{n}{\frac {I_{i}}{\pi _{i}}}=\sum _{i=1}^{n}{\check {1}}'_{i}}$. With the above notation, the parameter we care about is the ratio of the sums of ${\displaystyle y_{i}}$s, and 1s. I.e.: ${\displaystyle R={\bar {Y}}={\frac {\sum _{i=1}^{N}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}_{i}}{\sum _{i=1}^{N}{\check {1}}_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y_{i}}{\sum _{i=1}^{N}w_{i}}}}$. We can estimate it using our sample with: ${\displaystyle {\hat {R}}={\hat {\bar {Y}}}={\frac {\sum _{i=1}^{N}I_{i}{\frac {y_{i}}{\pi _{i}}}}{\sum _{i=1}^{N}I_{i}{\frac {1}{\pi _{i}}}}}={\frac {\sum _{i=1}^{N}{\check {y}}'_{i}}{\sum _{i=1}^{N}{\check {1}}'_{i}}}={\frac {\sum _{i=1}^{N}w_{i}y'_{i}}{\sum _{i=1}^{N}w_{i}1'_{i}}}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}={\bar {y}}_{w}}$. As we moved from using N to using n, we actually know that all the indicator variables get 1, so we could simply write: ${\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y_{i}}{\sum _{i=1}^{n}w_{i}}}}$. This will be the estimand for specific values of y and w, but the statistical properties comes when including the indicator variable ${\displaystyle {\bar {y}}_{w}={\frac {\sum _{i=1}^{n}w_{i}y'_{i}}{\sum _{i=1}^{n}w_{i}1'_{i}}}}$.