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In this question, you are being asked when an estimator \(\hat{\theta}\) of a parameter \(\theta\) is called **unbiased**.

An estimator \(\hat{\theta}\) is **unbiased** if the expected value of the estimator equals the true parameter value. This is mathematically defined as:

\[
E(\hat{\theta}) = \theta
\]

This can be rewritten as:

\[
E(\hat{\theta} - \theta) = 0
\]

Thus, an unbiased estimator satisfies this condition. Among the given options:

- (a) \(E(\hat{\theta} - \theta) = 0\) is the correct answer because this is the definition of an unbiased estimator.

The other options:
- (b) \(E(\hat{\theta} - \theta) = 1\) implies a bias of 1, so it is biased.
- (c) \(E(\hat{\theta} - \theta) > 1\) also implies a bias.
- (d) \(E(\hat{\theta} - \theta) < 1\) implies the estimator is biased as well.

Would you like further clarification on this concept or have any questions?

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Here are 5 related questions to deepen your understanding:

1. What is the difference between biased and unbiased estimators?
2. Can an estimator be consistent but biased?
3. How does bias affect the accuracy of an estimator?
4. What is the variance of an unbiased estimator?
5. How can you reduce the bias in an estimator?

**Tip**: A smaller bias doesn't always mean a better estimator; variance and overall mean squared error also play a role.

When is the estimator θ̂ of θ called unbiased?
(a) E(θ̂ − θ) = 0
(b) E(θ̂ − θ) = 1
(c) E(θ̂ − θ) > 1
(d) E(θ̂ − θ) < 1

Learn when an estimator θ̂ of parameter θ is called unbiased. This problem discusses key statistical concepts such as unbiased estimators and the condition E(θ̂ − θ) = 0. Suitable for undergraduate statistics students.

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