# MLE with General Optimization Functions in R

May 3, 2018
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In my previous post (https://statcompute.wordpress.com/2018/02/25/mle-in-r/), it is shown how to estimate the MLE based on the log likelihood function with the general-purpose optimization algorithm, e.g. optim(), and that the optimizer is more flexible and efficient than wrappers in statistical packages.

A benchmark comparison are given below showing the use case of other general optimizers commonly used in R, including optim(), nlm(), nlminb(), and ucminf(). Since these optimizers are normally designed to minimize the objective function, we need to add a minus (-) sign to the log likelihood function that we want to maximize, as shown in the minLL() function below. In addition, in order to speed up the optimization process, we can suppress the hessian in the function call. If indeed the hessian is required to calculate standard errors of estimated parameters, it can be calculated by calling the hessian() function in the numDeriv package.

As shown in the benchmark result, although the ucminf() is the most efficient optimization function, a hessian option can increase the computing time by 70%. In addition, in the second fastest nlminb() function, there is no built-in option to output the hessian. Therefore, sometimes it might be preferable to estimate model parameters first and then calculate the hessian afterwards for the analysis purpose, as demonstrated below.

```df <- read.csv("Downloads/credit_count.txt")

### DEFINE THE OBJECTIVE FUNCTION ###
minLL <- function(par) {
mu <- exp(par + par * df\$AGE + par * df\$ACADMOS + par * df\$MINORDRG + par * df\$OWNRENT)
return(ll <- -sum(log(exp(-mu) * (mu ^ df\$MAJORDRG) / factorial(df\$MAJORDRG))))
}

### BENCHMARKING ###
import::from("rbenchmark", "benchmark")
benchmark(replications = 10, order = "elapsed", relative = "elapsed",
columns = c("test", "replications", "elapsed", "relative"),
optim   = {optim(par = rep(0, 5), fn = minLL, hessian = F)},
nlm     = {nlm(f = minLL, p = rep(0, 5), hessian = F)},
nlminb  = {nlminb(start = rep(0, 5), objective = minLL)},
ucminf  = {ucminf::ucminf(par = rep(0, 5), fn = minLL, hessian = 0)},
hessian = {ucminf::ucminf(par = rep(0, 5), fn = minLL, hessian = 1)}
)
#      test replications elapsed relative
# 4  ucminf           10   4.044    1.000
# 3  nlminb           10   6.444    1.593
# 5 hessian           10   6.973    1.724
# 2     nlm           10   8.292    2.050
# 1   optim           10  12.027    2.974

### HOW TO CALCULATE THE HESSIAN ###
fit <- nlminb(start = rep(0, 5), objective = minLL)
import::from("numDeriv", "hessian")
std <- sqrt(diag(solve(hessian(minLL, fit\$par))))
est <- data.frame(beta = fit\$par, stder = std, z_values = fit\$par / std)
#           beta        stder   z_values
# 1 -1.379324501 0.0438155970 -31.480217
# 2  0.010394876 0.0013645030   7.618068
# 3  0.001532188 0.0001956843   7.829894
# 4  0.461129515 0.0068557359  67.261856
# 5 -0.199393808 0.0283222704  -7.040177

```

It is worth mentioning that, although these general optimizers are fast, they are less user-friendly than wrappers in statistical packages, such as mle or maxLik. For instance, we have to calculate AIC or BIC based on the log likelihood function or p-values based on Z-scores.

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