Expresses sample variance as a quadratic form

sample-variance-as-quadratic-form.R

n = 5

# Establish quadratic form

quad_form_matrix <- matrix(nrow = n, ncol = n)
for (i in seq(n)) {
  for (j in seq(n)) {
    if (i == j) {
      quad_form_matrix[i,j] <- (1/n)
    } else {
      quad_form_matrix[i,j] <- -1/((n*(n-1)))
    }
  }
}

# Calculate sample variance using usual formula ----

  ##_ Single variable
  y <- rnorm(n = n)
  var(y)
  t(y) %*% quad_form_matrix %*% y
  
  ##_ Multiple variables
  Y_mat <- MASS::mvrnorm(n = n, mu = c(1,5), Sigma = diag(2))
  t(Y_mat) %*% quad_form_matrix %*% Y_mat
  cov(Y_mat)