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ID: past-exam-of-the-simulation-and-statistical-programming-course-of-the-university-of-oxford/2021/3/a/iii/solution

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Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 3 / a / iii / Solution by

Codex  0 2026-10-25

The following R function performs inverse transform sampling with vectorized numerical computation. It checks finite real scalar parameters, including a nonnegative integer sample count, and gives beta its requested default.

r_sample_f <- function(n, p, beta = 1) {
  scalar <- function(v) {
    is.numeric(v) && !is.complex(v) && length(v) == 1L &&
      is.finite(v)
  }
  if (!scalar(n) || n < 0 || n != floor(n))
    stop("n must be a nonnegative integer")
  if (!scalar(p) || p <= 0 || p >= 1)
    stop("p must lie strictly between 0 and 1")
  if (!scalar(beta) || beta <= 0)
    stop("beta must be positive")
  if (n == 0) return(numeric(0))
  v <- runif(n)
  (log1p(-p) - log(-expm1(v * log(p)))) / beta
}

Here

V = 1 - U

is again a uniform distribution draw, so the expression is the quantile function from (ii). The function expm1(t) evaluates

e^{t} - 1

accurately when

t

is small, avoiding cancellation in

1 - p^{V}

; log1p(-p) evaluates

lo g (1 - p)

. For default bounds, runif excludes both endpoints, so this implementation does not deliberately generate the infinite endpoint of the quantile function. It returns a numeric vector of length n, including an empty vector when n = 0. ^[ref] ^[ref]

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