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I have two groups, Group 1 and Group 2, where the distribution of var is quite different. However, Group 1 has much more people than Group 2, and the support of Group 2 is a subset of the support of Group 1. That is, for every person in Group 2 with a given value of var, I would expect there to be a person with nearly the same value in Group 1. Here is an example:

require(ggplot2)
group1 <- rnorm(n = 8E4, mean =2, sd= 5)
group2 <- c ( rnorm(n = 5E3,mean =-2, sd= 0.5), rnorm(n=5E3, mean = 1, sd = 0.5))
label <- factor(c(rep("group1", 8E4), rep("group2", 1E4)))
var = c(group1,group2)
mydata <- data.frame(var,label)
ggplot(mydata, aes(x = var , fill=label))  + geom_histogram( position="identity", alpha=0.5 , bins = 100 )

before matching distributions

I would like to find a subset of Group 1 that has the same distribution of Group 2. I did so quite simply by using a simple histogram based density estimator, I estimate the density of the two distributions, and then sample from group 1 accordingly:

var_cut <- cut(var,breaks = 100)

#Simple density estimators
p_density<- table(var_cut[label == "group1"])/length(group1)
q_density<- table(var_cut[label == "group2"])/length(group2)

#Evaluate density of both distribution at each point of group 1 
p <- p_density[var_cut[label == "group1"]]
q <- q_density[var_cut[label=="group1"]]

matching_idx <- sample(which(label == "group1"),sum(label=="group2"), FALSE, q/p )

#plot it
mydata$matching_vec <- rep(NA, nrow(mydata))
mydata$matching_vec[matching_idx] <- "group1 (matching)"
mydata$matching_vec[label == "group2"] <- "group2"
mydata$matching_vec <- factor(mydata$matching_vec)

ggplot(subset(mydata, !is.na(matching_vec))) +  aes(x = var , fill=matching_vec)  + geom_histogram( position="identity", alpha=0.5 , bins = 100 )

after matching distributions

What is this called? Are there any theoretical results about this sort of process? Although it seemed to work pretty nicely, it is not perfect--I would love to better understand the theoretical implications of what empirically makes sense.

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I'm not sure if there's a better term, but what you are doing is known as "achieving balance" between the two groups on the covariate var. It is closely related to the technique of "matching," which is used often in causal inference in an attempt to make the treatment and control groups in an observational study look as similar as possible ("balanced") on all the other covariates. You have such a large population that this subsampling technique worked well to achieve balance; in matching, you just go and look for treatment-control pairs with similar values of the covariate(s). Here is a cool reference that should be accessible to folks with a technical undergrad degree.

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