Suppose we only have the following information:

• A set of sample means : $\bar{x_1}$, $\bar{x_2}$ ... $\bar{x_k}$
• The sample size used to calculated each sample mean: $n_1$, $n_2$ ....$n_k$
• The population size from which each sample mean was taken: $N_1$, $N_2$ ....$N_k$
• The sample variance of each mean: $Var(\bar{x_1})$, $Var(\bar{x_2})$ ... $Var(\bar{x_k})$
• We do NOT have the raw data that were used to calculate the sample means

Given this information, I have seen that this information can be combined together to produce a weighted mean estimator:

1) Count Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$ $$w_i = n_i$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

2) Variance Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$

$$w_i = \frac{1}{Var(\bar{x_i})}$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

Note that in this case, it can be shown (using Lagrange Multipliers) that this weighting scheme (i.e. inverse variance) will produce the estimate with the lowest overall variance (https://en.wikipedia.org/wiki/Inverse-variance_weighting)

My Question: Suppose we are given the exact same information (i.e. sample size, population size, sample variance, no raw data) - but this time, we are provided with the sample medians instead of the sample means.

Is it still common practice to calculate "weighted median" estimators just as we did above? Or does this "ruin" the purpose of the median by "contaminating" all the medians with each other (i.e. influencing) - thus no longer a "true median"? In general, I am not sure if it is a good idea to combine medians like this together: does this combined estimator still have desired mathematical properties (e.g. asymptotic normality, consistency).

I am not sure how exactly the variance of this weighted median estimator will be calculated. I am considering using the weighted bootstrap in this situation (i.e. the probability of selecting an individual median is proportional to its weight) - but I am not sure if this is a statistically valid approach (e.g. Understanding the Weighted Bootstrap)

References:

• The variance of the weighted median and optimal weights

Suppose we only have the following information:

• A set of sample means : $\bar{x_1}$, $\bar{x_2}$ ... $\bar{x_k}$
• The sample size used to calculated each sample mean: $n_1$, $n_2$ ....$n_k$
• The population size from which each sample mean was taken: $N_1$, $N_2$ ....$N_k$
• The sample variance of each mean: $Var(\bar{x_1})$, $Var(\bar{x_2})$ ... $Var(\bar{x_k})$
• We do NOT have the raw data that were used to calculate the sample means

Given this information, I have seen that this information can be combined together to produce a weighted mean estimator:

1) Count Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$ $$w_i = n_i$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

2) Variance Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$

$$w_i = \frac{1}{Var(\bar{x_i})}$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

Note that in this case, it can be shown (using Lagrange Multipliers) that this weighting scheme (i.e. inverse variance) will produce the estimate with the lowest overall variance (https://en.wikipedia.org/wiki/Inverse-variance_weighting)

My Question: Suppose we are given the exact same information (i.e. sample size, population size, sample variance, no raw data) - but this time, we are provided with the sample medians instead of the sample means.

Is it still common practice to calculate "weighted median" estimators just as we did above? Or does this "ruin" the purpose of the median by "contaminating" all the medians with each other (i.e. influencing) - thus no longer a "true median"? In general, I am not sure if it is a good idea to combine medians like this together: does this combined estimator still have desired mathematical properties (e.g. asymptotic normality, consistency).

I am not sure how exactly the variance of this weighted median estimator will be calculated. I am considering using the weighted bootstrap in this situation (i.e. the probability of selecting an individual median is proportional to its weight) - but I am not sure if this is a statistically valid approach (e.g. Understanding the Weighted Bootstrap)

References:

• The variance of the weighted median and optimal weights

Suppose we only have the following information:

• A set of sample means : $\bar{x_1}$, $\bar{x_2}$ ... $\bar{x_k}$
• The sample size used to calculated each sample mean: $n_1$, $n_2$ ....$n_k$
• The population size from which each sample mean was taken: $N_1$, $N_2$ ....$N_k$
• The sample variance of each mean: $Var(\bar{x_1})$, $Var(\bar{x_2})$ ... $Var(\bar{x_k})$
• We do NOT have the raw data used to calculate the sample means

Given this information, I have seen that this information can be combined together to produce a weighted mean estimator:

1) Count Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$ $$w_i = n_i$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

2) Variance Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$

$$w_i = \frac{1}{Var(\bar{x_i})}$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

Note that in this case, it can be shown (using Lagrange Multipliers) that this weighting scheme (i.e. inverse variance) will produce the estimate with the lowest overall variance (https://en.wikipedia.org/wiki/Inverse-variance_weighting)

My Question: Suppose we are given the exact same information (i.e. sample size, population size, sample variance, no raw data) - but this time, we are provided with the sample medians instead of the sample means.

Is it still common practice to calculate "weighted median" estimators just as we did above? Or does this "ruin" the purpose of the median by "contaminating" all the medians with each other (i.e. influencing) - thus no longer a "true median"? In general, I am not sure if it is a good idea to combine medians like this together: does this combined estimator still have desired mathematical properties (e.g. asymptotic normality, consistency).

I am not sure how exactly the variance of this weighted median estimator will be calculated. I am considering using the weighted bootstrap in this situation (i.e. the probability of selecting an individual median is proportional to its weight) - but I am not sure if this is a statistically valid approach (e.g. Understanding the Weighted Bootstrap)

References:

• The variance of the weighted median and optimal weights

Suppose we only have the following information:

• A set of sample means : $\bar{x_1}$, $\bar{x_2}$ ... $\bar{x_k}$
• The sample size used to calculated each sample mean: $n_1$, $n_2$ ....$n_k$
• The population size from which each sample mean was taken: $N_1$, $N_2$ ....$N_k$
• The sample variance of each mean: $Var(\bar{x_1})$, $Var(\bar{x_2})$ ... $Var(\bar{x_k})$
• We do NOT have the raw data that were used to calculate the sample means

Given this information, I have seen that this information can be combined together to produce a weighted mean estimator:

1) Count Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$ $$w_i = n_i$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

2) Variance Weighted Mean:

$$\bar{x_W} = \frac{\sum_{i=1}^{k} w_i \bar{x_i}}{\sum_{i=1}^{k} w_i}$$

$$w_i = \frac{1}{Var(\bar{x_i})}$$

$$Var(\bar{x_W}) = \frac{\sum_{i=1}^{k} w_i^2 Var(\bar{x_i})}{(\sum_{i=1}^{k} w_i)^2}$$

Note that in this case, it can be shown (using Lagrange Multipliers) that this weighting scheme (i.e. inverse variance) will produce the estimate with the lowest overall variance (https://en.wikipedia.org/wiki/Inverse-variance_weighting)

My Question: Suppose we are given the exact same information (i.e. sample size, population size, sample variance, no raw data) - but this time, we are provided with the sample medians instead of the sample means.

Is it still common practice to calculate "weighted median" estimators just as we did above? Or does this "ruin" the purpose of the median by "contaminating" all the medians with each other (i.e. influencing) - thus no longer a "true median"? In general, I am not sure if it is a good idea to combine medians like this together: does this combined estimator still have desired mathematical properties (e.g. asymptotic normality, consistency).

I am not sure how exactly the variance of this weighted median estimator will be calculated. I am considering using the weighted bootstrap in this situation (i.e. the probability of selecting an individual median is proportional to its weight) - but I am not sure if this is a statistically valid approach (e.g. Understanding the Weighted Bootstrap)

References:

• The variance of the weighted median and optimal weights