The MSE of an estimator is just the expected sum of squared differences, e.g., if the true parameter value is `x` and we have an estimator `x.hat`, then the MSE is ``` mse(x.hat) = E[(x.hat-x) vcov(x.hat) + bias(x.hat, x) ```
Usage
# S3 method for mle
mse(x, theta = NULL)Arguments
- x
the `mle` object to compute the MSE of.
- theta
true parameter value, defaults to `NULL` for unknown. If `NULL`, then we let the bias method deal with it. Maybe it has a nice way of estimating the bias.
Details
Since `x` is not typically known, we normally must estimate the bias. Asymptotically, assuming the regularity conditions, the bias of an MLE is zero, so we can estimate the MSE as `mse(x.hat) = vcov(x.hat)`, but for small samples, this is not generally the case. If we can estimate the bias, then we can replace the bias with an estimate of the bias.
Sometimes, we can estimate the bias analytically, but if not, we can use something like the bootstrap. For example, if we have a sample of size `n`, we can bootstrap the bias by sampling `n` observations with replacement, computing the MLE, and then computing the difference between the bootstrapped MLE and the MLE. We can repeat this process `B` times, and then average the differences to get an estimate of the bias.