如何为 R 中的不同 theta 绘制柯西分布对数似然的一阶导数图

Question

I have a set of observations from a Cauchy (theta,1) and I have a plot for the log-likelihood against different x values

obs=c(1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, -0.07, -1.05, -13.87, -2.53, -1.75, 0.27, 43.21)
ll_c=function(theta, obs){ #define Loglikelihood function for Cauchy(θ,1) distribution
  logl= sum(dcauchy(obs, location = theta, scale = 1, log = T)) 
  return(logl)
}
x  = seq(from=-10,to=10,by=0.1) #create test values 
ll = NULL
for (i in x){
  ll = c(ll, ll_c(i, obs)) #perform ll_c for all test values and store
}
plot(x, ll)

I also need to make a plot of the first derivative of the log-likelihood function against the same x values and I can not figure out how to do so.

fdll_c=function(theta,obs){
  Dlogl=D(sum(dcauchy(obs,location=theta,scale=1,log=T)),'theta')
  return(Dlogl)
}
fdll = NULL
for (j in x){
  fdll = c(fdll, fdll_c(j,obs))
}
plot(x,fdll)

I have tried different variations on this code, but every time it has come back with an error or with a derivative of 0 at all points.

Answer 1

Maybe the following answers the question.
It uses an explicit log-likelihood partial derivative function and then applies it to a vector around 0.

obs <- c(1.77, -0.23, 2.76, 3.80, 3.47, 56.75, -1.34, 4.24, -2.44, 3.29, 3.71, -2.40, 4.53, -0.07, -1.05, -13.87, -2.53, -1.75, 0.27, 43.21)

dll_theta <- function(x, theta, scale){
  cc <- (x - theta)/scale
  -2*sum(1/cc)/scale
}

x <- seq(from = -10, to = 10, by = 0.001)
y <- sapply(x, function(.x) dll_theta(obs, theta = .x, scale = 1))

i <- which(abs(y) > 1e15)
plot(x[-i], y[-i], pch = ".")