In this exercise, we will generate a 1000 random gamma deviates with its two parameters set to shape=20 and rate=2. We will then estimate the two parameters by using the mle2 function in the bbmle package. This function will also return the Z values, and the p-values. Note that we need to assume a distribution that we will use to fit the parameters (in general we will receive data, and we will need to assume which distribution is reasonable). In this case, since we are generating the data, we already know that the data comes from a gamma distribution.
We will use the bbmle package, which will allow us to maximize the log-likelihood. This package essentially wraps an appropriate numerical maximization routine; we only need to pass a function that computes the sum of the log-likelihood across the whole dataset.
- Generate 1000 random gamma deviations with its parameters set to shape=20 and rate=2 as follows:
library(bbmle)
N <- 1000
xx <- rgamma(N, shape=20, rate=2)
- Pass a function that computes the sum of the log-likelihood across the whole dataset as follows:
LL <- function(shape, rate) {
R = suppressWarnings(dgamma(xx, shape=shape, rate=rate))
return(-sum(log(R)))
}
- Estimate the two parameters by using the mle2 function in the bbmle package as follows:
P_1000 = mle2(LL, start = list(shape = 1, rate=1))
summary(P_1000)
The estimated coefficients, standard errors, and p-values (N=10) are as follows:
| Estimate | Std. error | Z value | p-value | |
| Shape | 19.04 | 0.84 | 22.54 | <2.2e-16*** |
| Rate | 1.89 | 0.08 | 22.68 | <2.2e-16*** |
The standard errors are very small relative to the estimated coefficients, which is to be expected as we have a large sample (1,000 observations). The p-values are consequently extremely small (the asterisks mark that these values are smaller than 0.001). When the p-values are small we say that they are significative (choosing a threshold is somewhat debatable, but most people use 0.05—in this case, we would say that they are highly significative).