An R implementation of https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4811631/.
This answers the question: the Wald test compares population medians,
but what about other parts of the distribution. Johnson and Romer
provided and algorithm to compare a limited set of percentiles between
two distributions. I created a function
`Wald_percentiles_test()`

to implement it, code is on Github.

```
source('wald_percentiles_test.R')
# population 1: bimodal
x1 <- c(rnorm(500, mean = 0, sd = 1), rnorm(500, 5, 1))
# pooulation 2: normal
# note: can be a different length!
x2 <- c(rnorm(800, mean = 1, sd = 1))
# percentiles to test
p <- c(0.25, 0.50, 0.75)
# number of bootstrap runs
n.B <- 500
Wald_percentiles_test(x1, x2, p, n.B)
```

```
## W > qchisq? TRUE * Differences btwn percentile profiles are statistically significant
## CI95 of diff at p= 0.25 : [ -0.67 , 0.12 ]
## CI95 of diff at p= 0.5 : [ -0.54 , 4.4 ]
## CI95 of diff at p= 0.75 : [ 3 , 3.8 ] * Significant difference
```