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--- re-building 'v00_geex_intro.Rmd' using rmarkdown


  |   SB1_estfun <- function(data){
  |     Y1 <- data$Y1
  |     function(theta){
  |         c(Y1 - theta[1],
  |          (Y1 - theta[1])^2 - theta[2])
  |     }
  |   } 


  |   library(geex)
  |   results <- m_estimate(
  |       estFUN = SB1_estfun, 
  |       data   = geexex,
  |       root_control = setup_root_control(start = c(1,1))) 


  |   n <- nrow(geexex)
  |   A <- diag(1, nrow = 2)
  |   
  |   B <- with(geexex, {
  |     Ybar <- mean(Y1)
  |     B11 <- mean( (Y1 - Ybar)^2 )
  |     B12 <- mean( (Y1 - Ybar) * ((Y1 - Ybar)^2 - B11) )
  |     B22 <- mean( ((Y1 - Ybar)^2 - B11)^2 )
  |     matrix(
  |       c(B11, B12,
  |         B12, B22), nrow = 2
  |     )
  |   })
  |   
  |   # closed form roots
  |   theta_cls <- c(mean(geexex$Y1),
  |                  # since var() divides by n - 1, not n
  |                  var(geexex$Y1) * (n - 1)/ n ) 
  |   
  |   # closed form sigma
  |   Sigma_cls <- (solve(A) %*% B %*% t(solve(A))) / n
  |   
  |   comparison <- list(geex = list(estimates = coef(results), vcov = vcov(results)), 
  |                      cls  = list(estimates = theta_cls, vcov = Sigma_cls)) 

  |  format(max(abs(comparison$geex$estimates - comparison$cls$estimates), abs(comparison$geex$vcov - comparison$cls$vcov)), digits = 1)  #757:891

  |   comparison 


  |   SB2_estfun <- function(data){
  |     Y1 <- data$Y1; Y2 <- data$Y2
  |     function(theta){
  |         c(Y1 - theta[1],
  |           Y2 - theta[2],
  |           theta[1] - (theta[3] * theta[2]) 
  |       )
  |     }
  |   } 


  |   results <- m_estimate(
  |       estFUN = SB2_estfun, 
  |       data  = geexex, 
  |       root_control = setup_root_control(start = c(1, 1, 1))) 


  |   # Comparison to an analytically derived sanwich estimator
  |   A <- with(geexex, {
  |    matrix(
  |     c(1 , 0, 0,
  |       0 , 1, 0,
  |       -1, mean(Y1)/mean(Y2), mean(Y2)),
  |     byrow = TRUE, nrow = 3)
  |   })
  |   
  |   B <- with(geexex, {
  |     matrix(
  |       c(var(Y1)   , cov(Y1, Y2), 0,
  |         cov(Y1, Y2), var(Y2)   , 0,
  |         0, 0, 0),
  |       byrow = TRUE, nrow = 3)
  |   })
  |   
  |   ## closed form roots
  |   theta_cls <- c(mean(geexex$Y1), mean(geexex$Y2))
  |   theta_cls[3] <- theta_cls[1]/theta_cls[2]
  |   ## closed form covariance
  |   Sigma_cls <- (solve(A) %*% B %*% t(solve(A))) / nrow(geexex)
  |   comparison <- list(geex = list(estimates = coef(results), vcov = vcov(results)), 
  |                      cls  = list(estimates = theta_cls, vcov = Sigma_cls)) 

  |  format(max(abs(comparison$geex$estimates - comparison$cls$estimates)), digits = 2)  #261:346
  |  format(max(abs(comparison$geex$vcov - comparison$cls$vcov)), scientific = TRUE, digits = 2)  #438:532

  |   comparison 


  |   SB3_estfun <- function(data){
  |     Y1 <- data$Y1
  |     function(theta){
  |         c(Y1 - theta[1],
  |          (Y1 - theta[1])^2 - theta[2],
  |          sqrt(theta[2]) - theta[3],
  |          log(theta[2]) - theta[4])
  |     }
  |   } 


  |   results <- m_estimate(
  |      estFUN= SB3_estfun, 
  |      data  = geexex,
  |      root_control = setup_root_control(start = rep(2, 4, 4, 4))) 


  |   ## closed form roots
  |   theta_cls <- numeric(4)
  |   theta_cls[1] <- mean(geexex$Y1)
  |   theta_cls[2] <- sum((geexex$Y1 - theta_cls[1])^2)/nrow(geexex)
  |   theta_cls[3] <- sqrt(theta_cls[2])
  |   theta_cls[4] <- log(theta_cls[2])
  |   
  |   ## Compare to closed form ##
  |   theta2 <- theta_cls[2]
  |   mu3 <- moments::moment(geexex$Y1, order = 3, central = TRUE)
  |   mu4 <- moments::moment(geexex$Y1, order = 4, central = TRUE)
  |   ## closed form covariance
  |   Sigma_cls <- matrix(
  |     c(theta2, mu3, mu3/(2*sqrt(theta2)), mu3/theta2,
  |       mu3, mu4 - theta2^2, (mu4 - theta2^2)/(2*sqrt(theta2)), (mu4 - theta2^2)/theta2,
  |       mu3/(2 * sqrt(theta2)), (mu4 - theta2^2)/(2*sqrt(theta2)), (mu4 - theta2^2)/(4*theta2), (mu4 - theta2^2)/(2*theta2^(3/2)),
  |       mu3/theta2, (mu4 - theta2^2)/theta2, (mu4 - theta2^2)/(2*theta2^(3/2)), (mu4/theta2^2) - 1) ,
  |     nrow = 4, byrow = TRUE) / nrow(geexex)
  |   ## closed form covariance
  |   # Sigma_cls <- (solve(A) %*% B %*% t(solve(A))) / n
  |   comparison <- list(geex = list(estimates = coef(results), vcov = vcov(results)), 
  |                      cls  = list(estimates = theta_cls, vcov = Sigma_cls)) 

  |  format(max(abs(comparison$geex$estimates - comparison$cls$estimates), abs(comparison$geex$vcov - comparison$cls$vcov)), digits = 2)  #234:368

  |   comparison 

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