8.1

1. Unbiasedness: all the bias result are around -1 to 1. The tauhat2 all had negative numbers which shows that it is a better predictor
2. Varaince: The variance for all n and predictors are all postive and under 10. Agian Tauhat2 have the lowest value number which again shows it is the better predictor when looking at the varaince.
3. MSE: Comparing the values show that tauhat2 again has the lowest values which tells us that it is the best predictor.

library(tidyverse)

Warning: package 'tidyverse' was built under R version 4.4.3

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Warning: package 'lubridate' was built under R version 4.4.3

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
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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

library(knitr)

Warning: package 'knitr' was built under R version 4.4.3

theta <- 5
true_median <- theta * log(2)

(parameters(~n,
            c(3,11,51)
)
  %>% add_trials(10000)
  %>% mutate(sample = map(n, ~rexp(.x, rate = 1/theta)))
  %>% mutate(ybar = map_dbl(sample, mean))
  %>% mutate(ymed = map_dbl(sample, median))
  %>% mutate(tauhat1 = ybar)
  %>% mutate(tauhat2 = log(2)*ybar)
  %>% mutate(tauhat3 = ymed)
  ) -> sim_data

summary_table <- 
sim_data %>%
    pivot_longer(tauhat1:tauhat3,
                 names_to = "estimator",
                 values_to = "estimate")%>%
  group_by(n, estimator) %>%
  summarize(
    bias = mean(estimate - true_median),
    variance = var(estimate),
    mse = bias^2 + variance,
    .groups="drop"
  )%>%
  mutate(across(c(bias, variance, mse), round, 3))

Warning: There was 1 warning in `mutate()`.
ℹ In argument: `across(c(bias, variance, mse), round, 3)`.
Caused by warning:
! The `...` argument of `across()` is deprecated as of dplyr 1.1.0.
Supply arguments directly to `.fns` through an anonymous function instead.

  # Previously
  across(a:b, mean, na.rm = TRUE)

  # Now
  across(a:b, \(x) mean(x, na.rm = TRUE))

kable(summary_table)

n	estimator	bias	variance	mse
3	tauhat1	1.496	8.427	10.664
3	tauhat2	-0.027	4.049	4.049
3	tauhat3	0.679	9.140	9.601
11	tauhat1	1.523	2.222	4.540
11	tauhat2	-0.008	1.067	1.068
11	tauhat3	0.198	2.315	2.354
51	tauhat1	1.535	0.492	2.849
51	tauhat2	0.001	0.236	0.236
51	tauhat3	0.048	0.495	0.497