CVEN9407 Chap.3 Estimators and Their Statistical Properties
Estimators and Their Statistical Properties
An estimator is a rule applied to a sample, so it has a sampling distribution of its own, and this chapter judges estimators by that distribution. It separates point from interval estimation, names the three estimation methods the course mentions, and defines bias, efficiency and mean square error. A worked example on spot speeds shows why dividing by n minus one gives an unbiased variance while maximum likelihood does not.
The chapter ends with the large-sample ideas of consistency, asymptotic efficiency and asymptotic normality.
These properties are the vocabulary used to defend a modelling choice.
When a report says least squares is the best linear unbiased estimator, that a coefficient is biased because of an omitted variable, or that a maximum likelihood estimate is reliable because the sample is large, it is using the language of this chapter. Markers and presentation questions often probe exactly those claims.
What this chapter covers
- 01
Estimators, estimates and sampling distributions
- 02
Point and interval estimation
- 03
Least squares, maximum likelihood and the method of moments
- 04
Unbiasedness, bias and mean square error
- 05
Best linear unbiased estimators
- 06
Consistency and asymptotic properties
Worked example · free
Bias of the divide-by-n variance on four spot speeds
- 1Mean = 184 / 4 = 46; squared deviations are 36, 4, 4 and 36, a total of 80.
- 1Dividing by n - 1 gives 80 / 3 = 26.67; dividing by n gives 80 / 4 = 20.
- 1The divide-by-n estimator has expectation (n - 1)/n times the variance, so with n = 4 it falls short by one quarter on average.
Key terms
- Estimator
- An estimator is a formula that turns sample observations into a value for an unknown population parameter.
- Bias
- Bias is the difference between an estimator's expected value and the true value of the parameter it estimates.
- Mean square error
- Mean square error is an estimator's variance plus its squared bias, the average squared distance from the true value.
- Consistent estimator
- A consistent estimator converges in probability to the true parameter value as the sample size grows without limit.
- Asymptotic efficiency
- Asymptotic efficiency is the property of having the smallest large-sample variance among all consistent estimators of a parameter.
Estimators and Their Statistical Properties FAQ
Can a biased estimator be the better choice?
Yes. If its variance is much smaller, its mean square error can be lower than that of an unbiased rival. The lectures note that minimum mean square error becomes the criterion when the best unbiased estimator still has too much variance.
What is the difference between unbiased and consistent?
Unbiased is a statement about the average of the estimator at a fixed sample size. Consistent is a statement about where it ends up as the sample grows. An estimator can be biased in every finite sample and still be consistent.
Why does maximum likelihood matter if the course mostly uses least squares?
Least squares suits the linear regression weeks, but the logit and survival models are estimated by maximum likelihood, so its logic and its small-sample behaviour return later in the term.
What does BLUE mean for the least squares slope?
Best linear unbiased estimator: among all estimators that are linear in the observations and unbiased, least squares has the smallest variance, provided the classical assumptions hold. When errors are heteroscedastic or autocorrelated, that guarantee no longer applies.
Why does a larger sample usually make estimates better?
Most estimators used in the course have variances that shrink as the sample grows, so their sampling distributions tighten around the true value. That is the practical meaning of consistency, and it is why pooling more years of crash data often sharpens results.
Assessment move
Learn the definitions as pairs of contrasts: point and interval, bias and variance, unbiased and consistent. Then rework the spot-speed example with your own five numbers and check that the two variance estimates differ by the factor n minus one over n. Finish by explaining aloud why the least squares slope earns the BLUE label.
Add one line to your model log for each estimator you use, saying which property justifies it for your sample size.
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