Explanation of the Unbiasedness Criterion of a Good Estimator

An estimator is unbiased if the expected value of the estimator equals the true population parameter it estimates. Mathematically, for an estimator $\hat{\theta}$ of a parameter $\theta$, it is unbiased if: $E[\hat{\theta}] = \theta$ This ensures that on average, across many samples, the estimator provides the correct parameter value.

However, in real-world scenarios, estimators may have some bias due to practical constraints. The bias of an estimator is defined as: $\text{Bias}(\hat{\theta}) = E[\hat{\theta}] - \theta$

In this problem, we compare two biased estimators of the same population parameter.

Comparison Between Estimators A and B

1. Bias of Estimator A:

   • Bias = $1/n$, which depends on the sample size $n$.
   • As $n$ increases, $1/n \to 0$, reducing the bias.

2. Bias of Estimator B:

   • Bias = 0.01, which is constant and independent of $n$.

When is Estimator A Better?

Estimator A becomes better than Estimator B under the condition that its bias $1/n$ is less than 0.01. Mathematically: $\frac{1}{n} < 0.01 \implies n > 100$

Thus, Estimator A is better than Estimator B when the sample size $n > 100$ because the bias of Estimator A becomes smaller than that of Estimator B.

Summary

• Estimator A: Bias decreases with larger sample size.
• Estimator B: Bias remains constant at 0.01.
• Conclusion: Estimator A is preferred when $n > 100$.

Do you want a deeper explanation or have specific questions?

Here are five related questions to expand your understanding:

1. What are the implications of bias in statistical estimation?
2. How does sample size influence the performance of an estimator?
3. Can you explain the trade-off between bias and variance in estimation?
4. Are there situations where a biased estimator is preferable to an unbiased one?
5. How is the Mean Squared Error (MSE) used to evaluate the overall performance of estimators?

Tip: Always consider both bias and variance when evaluating estimators; a trade-off often exists to minimize overall error.