Chapter 7.3

Question 4

library(tidyverse)
library(purrrfect)

(exponential_consec_stats_sim <- parameters(~i, ~n, ~lambda,
                                            c(3,5,9), c(10,15,20), c(0.05, 0.1)
                                            )
  %>% add_trials(10000)
  %>% mutate(y_sample = pmap(list(n, lambda), .f = \(n,l) rexp(n, rate = l)))
  %>% mutate(y_sorted = map(y_sample, sort))
  %>% mutate(yi = pmap_dbl(list(i, y_sorted), .f = \(i,y) pluck(y, i)),
             y_i_minus_1 = pmap_dbl(list(i, y_sorted), .f = \(i,y) pluck(y, i-1)),
             t = yi - y_i_minus_1)
  %>% mutate(f_t = dexp(t, lambda*(n-i+1)))
)

# A tibble: 180,000 × 10
       i     n lambda .trial y_sample   y_sorted    yi y_i_minus_1     t    f_t
   <dbl> <dbl>  <dbl>  <dbl> <list>     <list>   <dbl>       <dbl> <dbl>  <dbl>
 1     3    10   0.05      1 <dbl [10]> <dbl>     4.76       2.71  2.05  0.176 
 2     3    10   0.05      2 <dbl [10]> <dbl>     2.60       2.25  0.348 0.348 
 3     3    10   0.05      3 <dbl [10]> <dbl>     2.31       1.84  0.469 0.332 
 4     3    10   0.05      4 <dbl [10]> <dbl>     4.15       1.74  2.41  0.152 
 5     3    10   0.05      5 <dbl [10]> <dbl>     4.44       1.89  2.55  0.144 
 6     3    10   0.05      6 <dbl [10]> <dbl>     2.44       1.45  0.994 0.269 
 7     3    10   0.05      7 <dbl [10]> <dbl>    12.9       12.8   0.152 0.376 
 8     3    10   0.05      8 <dbl [10]> <dbl>     3.17       0.312 2.86  0.127 
 9     3    10   0.05      9 <dbl [10]> <dbl>     8.48       1.47  7.01  0.0243
10     3    10   0.05     10 <dbl [10]> <dbl>     7.59       3.23  4.36  0.0699
# ℹ 179,990 more rows

(ggplot(data = exponential_consec_stats_sim, aes(x = t))
 + geom_histogram(aes(y = after_stat(density)),
                  fill = 'goldenrod',
                  center = 0.1, binwidth = 0.2)
 + geom_line(aes(y = f_t), col = 'cornflowerblue')
 + facet_grid(lambda~i~n, labeller = label_both, scales = 'free_x')
 + coord_cartesian(xlim = c(0,5))
 + theme_classic()
 + labs(title = 'Simulated and analytic densities for consecutive order statistics of exponentials')
)