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Within a maximum likelihood context, the standard errors depend on the number of observations—the more observations we have, the smaller the standard errors will be (greater precision). As we can see in the following results, we get standard errors that are almost 50% of the estimated coefficients:

N <- 10
x <- rgamma(N, shape=20,rate=2)
LL <- function(shape, rate) {
  R = suppressWarnings(dgamma(x, shape=shape, rate=rate))
  return(-sum(log(R)))
}

P_10 = mle2(LL, start = list(shape = 1, rate=1))
summary(P_10)

The estimated coefficients and standard errors (N=10) are as follows:

The standard errors are much larger than before, almost 50% of their estimated coefficients. Consequently, the p-values are much larger than before, but still significative at the 0.05 level (which is why we get an asterisk). We still conclude that the coefficients are different from zero.

We can also compute confidence intervals using the confint function (in this case, we will use 95% intervals). These intervals can be inverted to get hypothesis tests. For example, we can test whether the shape is equal to 18 with a 95% confidence for our 1,000-sample example, by assessing if 18 is between the upper and lower boundaries; since 18 is between 17.30 and 20.59, we can't reject that the shape is equal to 18. Note that the confidence intervals are much tighter for the 1,000-sample case than for the 10-sample one. This is to be expected, as the precision depends on the sample size (we have already seen that the standard deviation for each estimated parameter depends on the sample size).

This is done via the following command:

confint(P_1000)
confint(P_10)

The confidence intervals are as follows: