All the tests were done on an Arch Linux x86_64 machine with an Intel(R) Core(TM) i7 CPU (1.90GHz). We first load the necessary packages.

library(melt)
library(microbenchmark)
library(ggplot2)

## Empirical likelihood computation

We show the performance of computing empirical likelihood with el_mean(). We test the computation speed with simulated data sets in two different settings: 1) the number of observations increases with the number of parameters fixed, and 2) the number of parameters increases with the number of observations fixed.

### Increasing the number of observations

We fix the number of parameters at $$p = 10$$, and simulate the parameter value and $$n \times p$$ matrices using rnorm(). In order to ensure convergence with a large $$n$$, we set a large threshold value using el_control().

set.seed(3175775)
p <- 10
par <- rnorm(p, sd = 0.1)
ctrl <- el_control(th = 1e+10)
result <- microbenchmark(
n1e2 = el_mean(matrix(rnorm(100 * p), ncol = p), par = par, control = ctrl),
n1e3 = el_mean(matrix(rnorm(1000 * p), ncol = p), par = par, control = ctrl),
n1e4 = el_mean(matrix(rnorm(10000 * p), ncol = p), par = par, control = ctrl),
n1e5 = el_mean(matrix(rnorm(100000 * p), ncol = p), par = par, control = ctrl)
)

Below are the results:

result
#> Unit: microseconds
#>  expr      min        lq       mean    median        uq      max neval
#>  n1e2    606.7    657.60    697.026    689.65    741.10    808.6   100
#>  n1e3   1715.4   1976.25   2115.239   2080.10   2239.05   3116.8   100
#>  n1e4  15952.3  18032.50  21109.374  21968.55  23012.85  29746.6   100
#>  n1e5 293123.7 368773.70 423891.477 413335.95 481057.45 565220.5   100
autoplot(result)

### Increasing the number of parameters

This time we fix the number of observations at $$n = 1000$$, and evaluate empirical likelihood at zero vectors of different sizes.

n <- 1000
result2 <- microbenchmark(
p5 = el_mean(matrix(rnorm(n * 5), ncol = 5),
par = rep(0, 5),
control = ctrl
),
p25 = el_mean(matrix(rnorm(n * 25), ncol = 25),
par = rep(0, 25),
control = ctrl
),
p100 = el_mean(matrix(rnorm(n * 100), ncol = 100),
par = rep(0, 100),
control = ctrl
),
p400 = el_mean(matrix(rnorm(n * 400), ncol = 400),
par = rep(0, 400),
control = ctrl
)
)
result2
#> Unit: milliseconds
#>  expr      min        lq       mean    median        uq      max neval
#>    p5   1.0614   1.11170   1.158171   1.16405   1.19390   1.2795   100
#>   p25   4.0403   4.08515   4.288955   4.13950   4.20765   9.2250   100
#>  p100  32.4020  32.66100  38.171283  35.84435  40.70450  61.7812   100
#>  p400 382.3951 412.21430 452.124979 421.48495 476.17470 673.0425   100
autoplot(result2)

On average, evaluating empirical likelihood with a 100000×10 or 1000×400 matrix at a parameter value satisfying the convex hull constraint takes less than a second.