Question

Some say that optimal estimators should be preferred while others advocate the use of more robust estimators. What is your opinion? (250 words)

Answer 1

In statistics, the term robust or robustness refers to the strength of a statistical model, tests, and procedures according to the specific conditions of the statistical analysis a study hopes to achieve. Given that these conditions of a study are met, the models can be verified to be true through the use of mathematical proofs.

However, many models are based upon ideal situations that do not exist when working with real-world data, and, as a result, the model may provide correct results even if the conditions are not met exactly.

Robust statistics, therefore, are any statistics that yield good performance when data is drawn from a wide range of probability distributions that are largely unaffected by outliers or small departures from model assumptions in a given dataset. In other words, a robust statistic is resistant to errors in the results.

One way to observe a commonly held robust statistical procedure, one needs to look no further than t-procedures, which use hypothesis tests to determine the most accurate statistical predictions.

Observing T-Procedures

For an example of robustness, we will consider t-procedures, which include the confidence interval for a population mean with unknown population standard deviation as well as hypothesis tests about the population mean.

The use of t-procedures assumes the following:

• The set of data that we are working with is a simple random sample of the population.
• The population that we have sampled from is normally distributed.

In practice with real-life examples, statisticians rarely have a population that is normally distributed, so the question instead becomes, “How robust are our t-procedures?”

In general the condition that we have a simple random sample is more important than the condition that we have sampled from a normally distributed population; the reason for this is that the central limit theorem ensures a sampling distribution that is approximately normal — the greater our sample size, the closer that the sampling distribution of the sample mean is to being normal.