Summarising simulations on a transformed parameter

Question

Let $\theta > 0$ be some model parameter for which properties (bias, ...) of an estimate are studied via simulations.

For a given data set, an estimate $\hat{\theta}_i$ of $\theta$ can be obtained by maximising a likelihood function.

As $\theta$ is positive, however, I perform the maximisation over $\eta = \log(\theta) \in \mathbb{R}$ and I set $\hat{\theta} = \exp(\hat{\eta})$.

After running $N$ simulations, how should I compute the mean value,

• $\bar{\hat{\theta}} = \frac{1}{N} \sum_{i=1}^N \hat{\theta}_i$, or
• $\bar{\hat{\theta}} = \exp(\frac{1}{N} \sum_{i=1}^N \hat{\eta}_i)$?

---

Example:

-- Minus log likelihood for a sample from the Exp distribution with mean $\theta = \exp(\eta)$:

minusloglik <- function(eta, sample)
{
  theta <- exp(eta)
  - sum(dexp(x=sample, rate=theta, log=TRUE))
}

-- True value of $\theta$:

theta <- 5.73

-- Simulations:

thetaHat <- etaHat <- rep(NA, 1000)
for(i in 1:1000)
{  
  sample <- rexp(n=100, rate=theta)
  etaHat[i] <- nlm(f=minusloglik, p=0, sample=sample)$estimate
  thetaHat[i] <- exp(etaHat[i])
}

Question:

• Should I summarise the results as mean(thetaHat) or as exp(mean(etaHat))?
• Is the answer the same if $\theta$ denotes the variance of a normal distribution?

Answer 1

If $\frac{1}{N} \sum_{i=1}^N \hat{\eta}_i$ is an unbiased estimate of the common $\eta$*, then exponentiating an unbiased estimate does not yield an unbiased estimate. If you are going to exponentiate the estimate of $\eta$, you need to account for that effect. Some people use Taylor expansions to do a first order correction, or assume a normal distribution for the estimate of $\eta$ and so might seek from that an unbiased estimate of $\exp(\eta)$.

* which it may not be, that depends on the distributional model and link function (and its correctness).

The same problem affects your other estimate - each of the individual estimates is biased, and averaging those biased estimates will also be biased.

In either case you'd need to adjust them.

Personally I wouldn't use either of those estimates anyway.

Consider that the individual estimates of $\eta$ are very unlikely to be equally uncertain. Then it makes no sense to give equal weight to a precise estimate and an imprecise estimate. Normally one would try to weight inversely proportional to variance, where possible.

(Consider an extreme case to illustrate the point. Imagine I have samples of size 10, 10, 10, 10, 10, and 10,000. If I take a plain average of the resulting five estimates of $\eta$, I'm much worse off - on average - than if I just throw away the first four!)

If you're interested in minimum mean square error, or some other property, you should perhaps look to optimize that, but you need to be clear whether the properties you seek are on the estimates of $\theta$ or $\log \theta$.

The properties of the various possible estimates of whatever quantity you're actually interested in can be investigated under various assumptions via simulation.

[The direct answers to your two final questions depend on things you are yet to specify. Please provide additional information.]