PS 8.3

Question 8

library(tidyverse)

Warning: package 'tidyverse' was built under R version 4.5.2

Warning: package 'ggplot2' was built under R version 4.5.2

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   4.0.1     ✔ tibble    3.3.0
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.1.0     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(purrrfect)

Attaching package: 'purrrfect'

The following objects are masked from 'package:base':

    replicate, tabulate

N <- 5000
epsilon <- 0.1

sim_results <- (
  parameters( ~n, ~theta,
              seq(50, 1000, by = 50), c(0.2, 0.5, 0.8)
  ) %>% 
    add_trials(N) %>% 
    mutate( 
      y_sample = pmap(list(n, theta), .f = \(n,t) rbeta(n, shape1 = t, shape2 = 3)),
      theta_hat = map_dbl(y_sample, \(y) 2 * mean(y / (1 - y))),
      within_eps = ifelse(abs(theta_hat - theta) < epsilon, 1, 0)
    ) %>% 
    summarize(
      prob_within_eps = mean(within_eps),
      .by = c(n, theta)
    )
)

ggplot(sim_results, aes(x = n, y = prob_within_eps, color = as.factor(theta))) +
  geom_line(linewidth = 1) +
  geom_hline(yintercept = 1, linetype = "dashed", color = "black", alpha = 0.5) +
  labs(
    title = "Simulated Convergence of Estimator",
    x = "n",
    y = expression(P(abs(hat(theta) - theta) < 0.1)),
    color = expression(True~theta)
  ) +
  scale_y_continuous(limits = c(0, 1.05), breaks = seq(0, 1, by = 0.2)) +
  theme_classic(base_size = 14)

As n grows, the prob converges to 1.