 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    533.2    589.80    629.985    634.35    666.90    739.9   100
#>  n1e3   1501.5   1723.90   1922.635   1857.80   2012.40   5412.7   100
#>  n1e4  14058.3  16234.25  19112.986  19934.20  21189.65  24134.5   100
#>  n1e5 276587.7 334928.05 395811.926 375773.60 472796.75 533855.2   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: microseconds
#>  expr      min        lq       mean    median        uq      max neval
#>    p5    889.2    973.40   1007.674   1010.65   1046.25   1124.4   100
#>   p25   3472.7   3569.30   3645.975   3651.90   3703.45   3950.1   100
#>  p100  30530.9  31440.25  37610.813  35058.50  39079.55  65164.9   100
#>  p400 370595.5 407161.15 452473.583 417586.80 479732.90 703250.3   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.