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Estimation of the Gamma Function

Estimation of the Gamma Function. Lecture XIXB. Estimate the gamma distribution using maximum likelihood:. Note that if we define two statistics. dta <- read.table ("Gamma Data 2007.dta") t1 <- sum( dta ) t2 <- sum(log( dta ))

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Estimation of the Gamma Function

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  1. Estimation of the Gamma Function Lecture XIXB

  2. Estimate the gamma distribution using maximum likelihood:

  3. Note that if we define two statistics

  4. dta <- read.table("Gamma Data 2007.dta") t1 <- sum(dta) t2 <- sum(log(dta)) fr <- function(b) { nrow(dta)*lgamma(b[1])+nrow(dta)*b[1]*log(b[2]) - (b[1]-1)*t2+1/b[2]*t1 } gfr <- function(b) { cbind(nrow(dta)*digamma(b[1])+nrow(dta)*log(b[2])-t2, nrow(dta)*b[1]/b[2]-1/b[2]^2*t1) } res <- optim(c(1.0,1.0),fr,gfr,method="BFGS") R-Code for Maximum Likelihood

  5. We know that the first four moments of the gamma distribution are Estimate the gamma distribution using method of moments.

  6. Actually the code minimized a squared error term

  7. dta <- read.table("Gamma Data 2007.dta") t1 <- mean(dta) t2 <- var(dta) fr <- function(b) { (b[1]*b[2] - t1)^2 + (b[1]*b[2]^2 - t2)^2 } gfr <- function(b) { cbind(2*(b[1]*b[2] - t1)*b[2] + 2*(b[1]*b[2]^2 - t2)*b[2]^2, 2*(b[1]*b[2] - t1)*b[1] + 4*(b[1]*b[2]^2 - t2)*b[1]*b[2]) } res <- optim(c(3.0,1.0),fr,gfr,method="BFGS") Exact Method of Moments

  8. Overidentified method of moments estimator

  9. dta <- read.table("Gamma Data 2007.dta") t1 <- mean(dta) t2 <- var(dta) t3 <- mean((dta-t1)^3) t4 <- mean((dta-t1)^4) fr <- function(b) { (b[1]*b[2] - t1)^2 + (b[1]*b[2]^2 - t2)^2 + (2/sqrt(b[1]) - t3)^2 + (6/b[1] - t4)^2 } gfr <- function(b) { cbind(2*(b[1]*b[2] - t1)*b[2] + 2*(b[1]*b[2]^2 - t2)*b[2]^2 -2*(2/sqrt(b[1]) - t3)/(b[1])^(3/2) - 12*(6/b[1] - t4)/b[1]^2, 2*(b[1]*b[2] - t1)*b[1] + 4*(b[1]*b[2]^2 - t2)*b[1]*b[2]) } res <- optim(c(3.0,2.0),fr,gfr,method="BFGS") Overidentified MOM

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