Choice of estimator

Generally, a population parameter can be estimated in a variety of different ways by using several different sample statistics. For example, the population mean can be estimated using estimators such as the sample mean, the sample median, and even more exotic sample statistics such as trimmed means etc.. This raises the question of which method to use to choose the best estimator. The three most frequently used estimation approaches are:

1. Moment method - the sample moments, $\overline{x}$, $\overline{x^2}$, $\overline{x^3}$, etc., are used to provide simple estimates of the location, scale, and shape parameters of the population distribution. Although these are the most intuitive choices for estimator, they have the disadvantage of giving biased estimates for non-normal distributions (non-robust), and can also be unduly influenced by the presence of outliers in the sample (non-resistant).

2. Robust estimation - instead of using moments, robust estimation methods generally use statistics based on quantiles e.g. median, interquartile range, L-moments, etc.. These measures are more robust and resistant but have the disadvantage of giving estimators that have larger sampling errors (i.e. less efficient estimators - see next section).

3. Maximum Likelihood Estimation (MLE) - These are most widely used estimators because of their many desirable properties. MLE estimates are parameter values chosen so as to maximise the likelihood of obtaining the data sample. In simple cases such as normally distributed data, the MLE procedure leads to moment estimates of the mean and variance. For more complicated cases, the MLE approach gives a clear and unambiguous approach for choosing the best estimator.