9.3 extra credit

For the joint log-likeihood plot below, it shows concavity in lambda and monotonicity in delta. For any fixed delta, the log-likeihood shows concavity in lambda. The horizontal slices show a single peak in the lamba direction which shows the vlaues decreasing on both sides of the plot. The presence of a unqiue maxium in lambda is shown in the plot. Monotonicity is shown in the graph below as delta increaes toward min(y). As delta approches min(y), the contours of the plot is being consistenly moving toward the higher values. The maxium of delta occurs at min(y) which is the largest allowable value of delta.

library(tidyverse)

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── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
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library(purrrfect)

Attaching package: 'purrrfect'

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

    replicate, tabulate

n <- 200

delta_true <- 2
lambda_true <- 1.5
mysamp <- delta_true + rexp(n, rate = lambda_true)

exp_loglik <- function(par, y) {
  lambda <- par[1]
  delta <- par[2]
  
  if(delta >= min(y) || lambda <= 0) return(-Inf)
  
  n <- length(y)
  n*log(lambda) - lambda * sum(y - delta)
}

lambdseq <- seq(0.5, 3, length.out = 100)
deltaseq <- seq(min(mysamp) - 1, min(mysamp) - 1e-3, length.out = 100)

loglik_grid <- expand.grid(lambda = lambdseq, delta = deltaseq) %>%
  as_tibble()%>%
  mutate(loglik = pmap_dbl(list(lambda, delta),
                           \(l,d) exp_loglik(c(l,d), mysamp)
                           ))

loglik_grid %>%
  slice_max(loglik, n = 1)

# A tibble: 1 × 3
  lambda delta loglik
   <dbl> <dbl>  <dbl>
1   1.36  2.00  -137.

ggplot(loglik_grid, aes(x = lambda, y = delta, z = loglik))+
  geom_contour_filled()+
  geom_contour(color = "grey") + 
  labs(x = expression(lambda), y = expression(delta)) + 
  theme_classic(base_size = 18)

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