How to find the second derivative in R and while using newton's method with numerical derivation

Question

The log-likelihood of the gamma distribution with scale parameter 1 can be written as:

(α−1)s−nlogΓ(α) where alpha is the shape parameter and s=∑logXi is the sufficient statistic.

Randomly draw a sample of n = 30 with a shape parameter of alpha = 4.5. Using newton_search and make_derivative , find the maximum likelihood estimate of alpha. Use the moment estimator of alpha, ie, mean of x as the initial guess. The log-likelihood function in R is:

x <- rgamma(n=30, shape=4.5)
gllik <- function() {
  s <- sum(log(x))
  n <- length(x)
  function(a) {
    (a - 1) * s - n * lgamma(a)
    }
}

I have created the make_derivative function as follows:

make_derivative <- function(f, h) {
  (f(x + h) - f(x - h)) / (2*h)
}

I also have created a newton_search function that incorporates the make_derivative function; However, I need to use newton_search on the second derivative of the log-likelihood function and I'm not sure how to fix the following code in order for it to do that:

newton_search2 <- function(f, h, guess, conv=0.001) {
    set.seed(2)  
    y0 <- guess
    N = 1000
    i <- 1; y1 <- y0
    p <- numeric(N)
  while (i <= N) {
    make_derivative <- function(f, h) {
  (f(y0 + h) - f(y0 - h)) / (2*h)
    }
    y1 <- (y0 - (f(y0)/make_derivative(f, h)))
    p[i] <- y1
    i <- i + 1
    if (abs(y1 - y0) < conv) break
    y0 <- y1
  }
  return (p[(i-1)])
}

Hint: You must apply newton_search to the first and second derivatives (derived numerically using make_derivative ) of the log-likelihood. Your answer should be near 4.5.

when I run newton_search2(gllik(), 0.0001, mean(x), conv = 0.001) , I get double what the answer should be.

Answer 1

I re-wrote the code and it works perfectly now (even better than what I had originally wrote). Thanks to all who helped. :-)

newton_search <- function(f, df, guess, conv=0.001) {
    set.seed(1)
    y0 <- guess
    N = 100
    i <- 1; y1 <- y0
    p <- numeric(N)
  while (i <= N) {
    y1 <- (y0 - (f(y0)/df(y0)))
    p[i] <- y1
    i <- i + 1
    if (abs(y1 - y0) < conv) break
    y0 <- y1
  }
  return (p[(i-1)])
}

make_derivative <- function(f, h) {
  function(x){(f(x + h) - f(x - h)) / (2*h)
  }
}

df1 <- make_derivative(gllik(), 0.0001)
df2 <- make_derivative(df1, 0.0001)
newton_search(df1, df2, mean(x), conv = 0.001)