Section 10.7 Summary
Subsection 10.7.1 Glossary
- Point Estimation
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An attempt to obtain the best guess of the value of a population parameter. An estimator is a statistic that produces such a guess. The estimate is the observed value of the estimator.
- Bias
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The difference between the expectation of the estimator and the value of the parameter. An estimator is unbiased if the bias is equal to zero. Otherwise, it is biased.
- Mean Square Error (MSE)
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A measure of the concentration of the distribution of the estimator about the value of the parameter. The mean square error of an estimator is equal to the sum of the variance and the square of the bias. If the estimator is unbiased then the mean square error is equal to the variance.
- Bernoulli Random Variable
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A random variable that obtains the value “1” with probability \(p\) and the value “0” with probability \(1-p\text{.}\) It coincides with the \(\mathrm{Binomial}(1,p)\) distribution. Frequently, the Bernoulli random variable emerges as the indicator of the occurrence of an event.
Subsection 10.7.2 Discuss in the forum
Performance of estimators is assessed in the context of a theoretical model for the sampling distribution of the observations. Given a criteria for optimality, an optimal estimator is an estimator that performs better than any other estimator with respect to that criteria. A robust estimator, on the other hand, is an estimator that is not sensitive to misspecification of the theoretical model. Hence, a robust estimator may be somewhat inferior to an optimal estimator in the context of an assumed model. However, if in actuality the assumed model is not a good description of reality then robust estimator will tend to perform better than the estimator denoted optimal.
Some say that optimal estimators should be preferred while other advocate the use of more robust estimators. What is your opinion?
When you formulate your answer to this question it may be useful to come up with an example from you own field of interest. Think of an estimation problem and possible estimators that can be used in the context of this problem. Try to identify a model that is natural to this problem an ask yourself in what ways may this model err in its attempt to describe the real situation in the estimation problem.
As an example consider estimation of the expectation of a Uniform measurement. We demonstrated that the mid-range estimator is better than the sample average if indeed the measurements emerge from the Uniform distribution. However, if the modeling assumption is wrong then this may no longer be the case. If the distribution of the measurement in actuality is not symmetric or if the distribution is more concentrated in the center than in the tails then the performance of the mid-range estimator may deteriorate. The sample average, on the other hand is not sensitive to the distribution not being symmetric.
Subsection 10.7.3 Formulas
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Bias: \(\mathrm{Bias} = \Expec(\hat \theta) - \theta\text{.}\)
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Variance: \(\Var(\hat \theta) = \Expec\big[(\hat \theta - \Expec(\hat\theta))^2\big]\text{.}\)
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Mean Square Error: \(\mathrm{MSE} = \Expec\big[(\hat \theta - \theta)^2\big]\text{.}\)
