Consider two estimators of
𝜃
.

𝜃
^
1
has expected value
𝜃
and variance
𝜃
𝑛
.

𝜃
^
2
has expected value
𝜃
and variance
𝜃
𝑛
2
.

What is the bias of
𝜃
^
1
?

Question

Consider two estimators of 
𝜃
.

𝜃
^
1
has expected value 
𝜃
 and variance 
𝜃
𝑛
.

𝜃
^
2
has expected value 
𝜃
 and variance 
𝜃
𝑛
2
.

What is the bias of 
𝜃
^
1
?

BlackTom AI · Accepted Answer

Bias of an estimator is defined as E[θ̂] − θ.

1. Relevant knowledge points and concepts
- The bias of an estimator θ̂ is bias(θ̂) = E[θ̂] − θ. If this equals 0 the estimator is unbiased.
- Expectation (mean) and variance are different characteristics: expectation enters bias; variance does not directly change bias.
- Given: for θ̂1, E[θ̂1] = θ and Var(θ̂1) = θ^n. The variance value does not affect the bias calculation except to remind us variability is present.

2. Complete reasoning process step by step
- Start with the definition: bias(θ̂1) = E[θ̂1] − θ.
- Substitute the given expected value: E[θ̂1] = θ, so bias(θ̂1) = θ − θ.

3. Intermediate calculation steps
- bias(θ̂1) = E[θ̂1] − θ
- = θ − θ
- = 0

4. Why the correct answer is correct
- Because the expected value of θ̂1 equals the true parameter θ, the estimator's mean deviation from θ is zero. That is precisely the definition of an unbiased estimator. The provided variance θ^n only quantifies dispersion around the mean and has no effect on the bias computation.

5. Why the other options are incorrect
- Option "θ": would assert E[θ̂1] − θ = θ, which implies E[θ̂1] = 2θ; this contradicts the given E[θ̂1] = θ.
- Option "θ^n": is the variance of θ̂1, not the bias. Variance ≠ E[θ̂] − θ in general.
- Option "θ^2 + θ^n": is an arbitrary combination of moments; there is no basis for this expression from the definition of bias and the given information.

6. Conclusion
- The bias of θ̂1 is 0, matching the provided correct answer.

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