AccurateVarianceEstimation/backFuncs.R at main · deepbas/AccurateVarianceEstimation · GitHub

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## -----------------------------------------------------
# Function to simulate VAR model parameters
# for different dependence/correlation

sigphi <- function(p, rho)
{
  set.seed(16235, kind = "L'Ecuyer-CMRG")
  A <- matrix(rnorm(p*p, mean = 0, sd = 1), p, p)
  B <- A %*% t(A)
  m <- max(eigen(B)$values)
  phi0 <- B/(m + 0.001)
  phik <- bdiag(rho*phi0)

  scratch <- diag((p)^2) - kronecker(phik, phik)
  V.s <- solve(scratch)%*%c(diag(p))
  V <- matrix(V.s, nrow = p, byrow = TRUE)
  Sigma <- solve(diag(p) - phik) %*% V + V %*% solve(diag(p) - phik) -V
  return(list(phi = phik, Sigma = Sigma, V = V))
}


## -----------------------------------------------------
# Function simulates an AR(1) model

ar1 <- function(N, phi, omega, start)
{
  out <- numeric(length = N)
  out[1] <- start
  eps <- rnorm(N, sd = sqrt(omega))
  for(t in 2:N)
  {
    out[t] <- phi*out[t-1] + eps[t]
  }
  return(out)
}

## -----------------------------------------------------
# Function that calculates true autocov for VAR(1)
true_autocov <- function(phi, V, lag)
{
  # to calculate the powers of phi
  foo <- svd(phi)
  d <- foo$d

  out <- matrix(0, nrow = lag+1, ncol = length(d))
  for(k in 0:lag)
  {
    out[k+1, ] <- diag(foo$u %*% diag(d^k) %*% t(foo$v) %*% V)
  }

  return(out)
}


## -----------------------------------------------------
# Function to calculate the exact bias and variance

funBMexact <- function(b, x, y, r, c){
  n <- length(x)
  b <- floor(b)
  br <- floor(b/r)
  a <- n/b
  ar <- n/br

  gamma0b1  <- 2*sum(x[2:b])
  gamma0b1r <- 2*sum(x[2:br])
  gamma0n1  <- 2*sum(x[(b+1):n])
  gamma0n1r <- 2*sum(x[(br+1):n])

  gamma1b1  <- 2*sum((1:(b-1))*x[2:b])
  gamma1b1r <- 2*sum((1:(br-1))*x[2:br])
  gamma1n1  <- 2*sum(x[(b+1):n]*(b:(n-1)))
  gamma1n1r <- 2*sum(x[(br+1):n]*(br:(n-1)))

  bias1     <- x[1] + gamma0b1 - (a + 1)/(a*b)*gamma1b1 -
                1/(a - 1)*(gamma0n1  -  gamma1n1/n )
  bias2     <- x[1] + gamma0b1r - (ar + 1)/(ar*br)*gamma1b1r -
                 1/(ar - 1)*(gamma0n1r  -  gamma1n1r/n )

  bias.exact      <- y - ( (1/(1-c))*bias1 - (c/(1-c))*bias2)

  variance.exact  <- (2*y^2*b/n)/(1 - c)^2 * (1 + b/(n-b) +
                      c*(c-2)*( (1/r) + b/(r*(n*r - b))) )
  mse.exact       <- bias.exact^2 + variance.exact
  return(mse.exact)
}


## -----------------------------------------------------
# Function to calculate the proposed bias and variance for BM estimators

funBMi <- function(b, x, y, r, c){
  n <- length(x)
  b <- floor(b)
  br <- floor(b/r)
  gamma0b1  <- 2*sum(x[2:b])
  gamma0b1r <- 2*sum(x[2:br])
  gamma0n1  <- 2*sum(x[2:n])

  gamma1b1  <- 2*sum((1:(b-1))*x[2:b])
  gamma1b1r <- 2*sum((1:(br-1))*x[2:br])
  gamma1n1  <- 2*sum((1:(n-1))*x[2:n])

  bias1     <- - ((n/(n-b)) *(gamma0n1 - gamma0b1 + gamma1b1/b + (b*gamma1n1)/(n^2)) )
  bias2     <- -((n/(n-br))*(gamma0n1 - gamma0b1r + gamma1b1r/br + (br*gamma1n1)/(n^2)))
  bias      <- (1/(1-c))* bias1 - (c/(1-c))* bias2
  variance       <- (2*y^2*b/n)/(1 - c)^2 * (1 + b/(n-b) +
                  c*(c-2)*( (1/r) + b/(r*(n*r - b))) )
  mse.bm    <- bias^2 + variance
  return(mse.bm)
}


## -----------------------------------------------------
# Function to calculate the proposed bias and variance for OBM estimators

# funOBMi <- function(b, x, y, r, c){
#   b <- floor(b)
#   n <- length(x)
#   br <- b/r
#   gamma0b1 <- 2*sum(x[2:b])
#   gamma0nb <- 2*sum(x[2:(n-b+1)])
#   gamma0nbr <- 2*sum(x[2:(n-br+1)])
#   gamma0b1r <- 2*sum(x[2:br])
#   gamma0n1 <- 2*sum(x[2:n])
#   gamma1b1 <- 2*sum((1:(b-1))*x[2:b])
#   gamma1b1r <- 2*sum((1:(br-1))*x[2:br])
#   bias <- (1/(1-c))*(gamma0n1 - gamma0b1 + (n*gamma1b1)/(b*(n-b)) -
#                        (b*n)/((n-b)*(n-b+1))*(gamma0b1 - gamma0nb)) -
#     (c/(1-c)*(gamma0n1 - gamma0b1r + (n*gamma1b1r)/(br*(n-br)) -
#                 (br*n)/((n-br)*(n-br+1))*(gamma0b1r - gamma0nbr)))
#   var <- ((4/3)*(y^2)*b/n) * (1/r + (r-1)/(r*(1-c)^2)) +
#     ((4/3)*(y^2)*(b/n)^2) * ((2*c)/((1-c)^2*r^2) - 2/r)
#   mse.obm <- bias^2 + var
#   return(mse.obm)
# }


## -----------------------------------------------------
# current
funCurrbm <- function(b, n, x, y, r, c)
{
  b <- floor(b)
  a <- n/b
  mse.bm.curr <-  ((x*(a+1)/n)*(1-r*c)/(1-c))^2 +
                  2*b/n*(y^2)*(1/r + (r-1)/(r*(1-c)^2))
                  # mse.bm.curr <- na.exclude(mse.bm.curr)
  return(mse.bm.curr)
}

# funCurrobm <- function(b, x, y, r, c){
#   mse.obm.curr <- ((x/b)*(1-r*c)/(1-c))^2 +
#                   ((4/3)*(y^2)*b/n) * (1/r + (r-1)/(r*(1-c)^2)) +
#                   ((4/3)*(y^2)*(b/n)^2) * ((2*c)/((1-c)^2*r^2) - 2/r)
#   mse.obm.curr <- na.exclude(mse.obm.curr)
#   return(mse.obm.curr)
# }