ETC3400 Chap.2 Estimators, Sampling Distributions and Bias
Estimators, Sampling Distributions and Bias
Estimators, Sampling Distributions and Bias separates the rule used before data arrive from the numerical result produced after observation. An estimator maps every possible sample to a parameter value and is therefore a random variable with its own probability distribution.
The chapter builds an exact Bernoulli sampling distribution by enumeration, then uses expectation and variance to define bias, precision and mean squared error. Jensen reasoning shows why nonlinear transformations can introduce bias even when the underlying moment is unbiased. The goal is to compare statistical rules under repeated sampling, not judge an estimator from one convenient dataset.
What this chapter covers
- 01
Function notation, domain and codomain for an estimator
- 02
Estimator based on random sample versus estimate based on observed data
- 03
Sampling distribution created by mapping every possible sample
- 04
Multiplicity when several datasets produce the same estimate
- 05
Repeated sample expectation and the definition of bias
- 06
Sampling variance as a measure of dispersion
- 07
Mean squared error equals variance plus squared bias
- 08
Jensen inequality for transformed estimators
Worked example · free
Build the sampling distribution of a sample proportion
- +1The sample proportion can equal 0, one third, two thirds or 1.
- +3The corresponding probabilities are (1-p) cubed, 3p(1-p) squared, 3p squared (1-p), and p cubed.
- +1The coefficients count the distinct binary sequences that map to each proportion and the four probabilities sum to 1.
- +2Multiplying values by probabilities and summing gives E[p hat] equal to p.
- +1Therefore the sample proportion is unbiased for every admissible p, not just at one selected value.
Key terms
- Estimator
- A function of a random sample that assigns one parameter value to every possible dataset and therefore has a sampling distribution.
- Estimate
- The fixed numerical value obtained when the estimator rule is applied to the sample that was actually observed.
- Sampling distribution
- The probability law of an estimator across repeated samples drawn from the stated model.
- Bias
- The repeated sample expectation of an estimator minus its target parameter. Unbiasedness requires zero bias throughout the parameter space.
- Mean squared error
- Expected squared estimation error, equal to variance plus squared bias when the target is fixed.
- Jensen inequality
- A relation connecting a transformation of an expectation with the expectation of a transformed random variable, with direction determined by convexity or concavity.
Estimators, Sampling Distributions and Bias FAQ
What is the difference between an estimator and an estimate?
An estimator is a random rule applied to all possible samples, while an estimate is the fixed value obtained from one realised dataset. Expectations, variances and sampling distributions belong to the estimator.
Can a biased estimator still be useful?
Yes. A biased estimator can be consistent and can have lower mean squared error if its reduction in variance outweighs the squared bias. The comparison criterion must be stated rather than assumed.
Why do repeated sample probabilities depend on the true parameter?
The statistical model assigns different probabilities to possible datasets as the true parameter changes. The estimator rule stays fixed, but its output distribution inherits that parameter dependence.
How does Jensen inequality reveal the direction of bias?
Identify whether the transformation is concave or convex, apply the inequality in the correct direction, then substitute the known expectation. Saying only that nonlinearity creates bias does not determine whether it is upward or downward.
Exam move
Draw a two column line on every problem: random notation on the left and observed notation on the right. Use capital letters for the estimator input and lower case for the realised dataset. For small discrete models, enumerate the exact sampling distribution and confirm its probabilities total one before calculating moments.
When comparing estimators, report bias, variance and mean squared error separately so the tradeoff remains visible. Practise one Jensen argument with a concave transformation and another with a convex transformation, writing the inequality direction before simplifying. Use simulation only as a reproducible check of a theoretical result, with a recorded seed and enough replications to discuss Monte Carlo error.
Turn each tutorial example into a four-row audit: estimator rule, possible outputs, output probabilities and target parameter. This makes multiplicities and repeated-sample reasoning explicit. For comparison questions, choose two parameter values and two sample sizes, calculate all three criteria, and identify whether the ranking changes. If it does, describe the region instead of declaring one estimator universally superior.
In simulation work, print theoretical and empirical probabilities side by side and explain whether discrepancies are compatible with replication noise. Revisit the same task without code and prove the relevant expectation algebraically. End each response with a sentence that distinguishes a property of the procedure from the numerical estimate produced by the observed data.
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