MGF Method Verification Practice

Problem 1

Show that if \(Y\sim POI(\lambda)\), that \(U = cY\) is NOT \(POI(c\lambda)\).

Show for a grid of \(\lambda \in \{1,2,3,4\}\) and with \(c = 2\) that \(U\) is NOT \(POI(2\lambda)\).

library(tidyverse)
library(purrrfect)

N <- 10000
(scaled_pois_sims <- parameters(~lambda, 
                                c(1,2,3,4))
  %>% add_trials(N)
  %>% mutate(Y = map_int(lambda, .f = \(l) rpois(1, l)))
  %>% mutate(U = 2*Y)
  %>% mutate(Fhat = cume_dist(U), .by = lambda,
             F = ppois(U, lambda = 2*lambda))
) %>% head

# A tibble: 6 × 6
  lambda .trial     Y     U  Fhat     F
   <dbl>  <dbl> <int> <dbl> <dbl> <dbl>
1      1      1     0     0 0.367 0.135
2      1      2     0     0 0.367 0.135
3      1      3     1     2 0.731 0.677
4      1      4     0     0 0.367 0.135
5      1      5     0     0 0.367 0.135
6      1      6     0     0 0.367 0.135

(ggplot(data = scaled_pois_sims)
  + geom_step(aes(x = U, y = Fhat, col = 'Empirical'))
  + geom_step(aes(x = U, y = F, col = 'Analytic'))
  + facet_wrap(~lambda)
  + theme_classic()
)

(ggplot(data = scaled_pois_sims)
  + geom_point(aes(x = F, y = Fhat))
  + geom_abline(aes(intercept = 0, slope = 1))
  + facet_wrap(~lambda, labeller = label_both)
)

Problem 2

If \(X \sim \chi^2_p\) and \(Y \sim \chi^2_q\), show that \(U = X + Y \sim \chi^2_{p+q}\) for \(p, q \in \{1,2,3,4\}\).

(chi_sims <- parameters(~p, ~q,
                        1:4, 1:4)
 %>% add_trials(N)
 %>% mutate(X = map_dbl(p, .f = \(p) rchisq(1, df = p)),
            Y = map_dbl(q, .f = \(q) rchisq(1, df = q)))
 %>% mutate(U = X + Y)
 %>% mutate(fU = dchisq(U, df = p+q))
) %>% head

# A tibble: 6 × 7
      p     q .trial      X        Y     U     fU
  <int> <int>  <dbl>  <dbl>    <dbl> <dbl>  <dbl>
1     1     1      1 0.157  0.0311   0.188 0.455 
2     1     1      2 2.22   0.309    2.53  0.141 
3     1     1      3 0.204  0.182    0.386 0.412 
4     1     1      4 5.49   1.34     6.84  0.0164
5     1     1      5 1.59   0.000286 1.59  0.225 
6     1     1      6 0.0602 0.559    0.619 0.367 

(ggplot(data = chi_sims)
  + geom_histogram(aes(x = U, y = after_stat(density)))
  + geom_line(aes(x = U, y = fU), col = 'cornflowerblue')
  + facet_grid(p~q, labeller = label_both)
  + theme_classic()
)

`stat_bin()` using `bins = 30`. Pick better value `binwidth`.