Definition
An unbiased estimator is a statistical method used to estimate a population parameter in such a way that the expected value of the estimate equals the true value of the population parameter. In other words, the method is considered unbiased if, when applied to many random samples from the population, the average of the results equals the actual population parameter.
Examples
- Credit Card Balances: Consider a scenario where you are tasked with estimating the average account balance of credit card holders in a city. If multiple random samples are taken from the entire city’s credit card holders, and each sample’s average balance is calculated, the mean of these averages should equal the actual average balance of all account holders in the city for the estimator to be considered unbiased.
- Heights of Students: Suppose you want to estimate the average height of students at a university. If you were to choose several random samples of students over a period, and the average height from these samples matches the real average height of all university students, your estimator is unbiased.
Frequently Asked Questions (FAQs)
Q1: What makes an estimator unbiased?
A1: An estimator is unbiased if the expected value (mean) of its sampling distribution is equal to the true value of the parameter being estimated.
Q2: Why is unbiasedness important?
A2: Unbiasedness ensures that, on average, our estimates are correct and not systematically off, which increases the reliability and accuracy of the inference we make about the population.
Q3: Can an estimator be both unbiased and inconsistent?
A3: Yes, it is possible for an estimator to be unbiased but inconsistent. An unbiased estimator might not converge to the true parameter value as sample size increases (lack of consistency).
Q4: How can we test if an estimator is unbiased?
A4: To test if an estimator is unbiased, one can compute the expected value of the estimator and verify if it equals the true parameter.
Q5: Are there situations where a biased estimator is preferred?
A5: Yes, in some cases, biased estimators are preferred if they offer significantly lower variance or mean squared error compared to unbiased estimators, making them more stable or reliable in practical applications.
- Estimator: A statistic used to estimate the value of a population parameter.
- Biased Estimator: An estimator with a systematic error, where the expected value of the estimate does not equal the true population parameter.
- Sampling Distribution: The probability distribution of a given statistic based on a random sample.
- Population Parameter: A value that quantitatively describes a characteristic of a population.
Online References
Suggested Books for Further Studies
- “Statistical Inference” by George Casella and Roger L. Berger
- “Introduction to the Theory of Statistics” by Alexander M. Mood, Franklin A. Graybill, and Duane C. Boes
Fundamentals of Unbiased Estimator: Statistics Basics Quiz
### What is an unbiased estimator?
- [ ] An estimator that consistently overestimates the population parameter.
- [x] An estimator whose expected value is equal to the population parameter.
- [ ] An estimator that produces the same result regardless of the sample.
- [ ] An estimator with the smallest possible variance.
> **Explanation:** An unbiased estimator is one whose expected value or average of several sample estimates is equal to the population parameter.
### Which of the following describes an unbiased sample?
- [ ] A sample taken from a specific section of the population.
- [x] A sample taken randomly from the entire population.
- [ ] A sample where each subgroup is equally represented.
- [ ] A sample with the highest number of observations.
> **Explanation:** For a sample to be unbiased, it must be randomly taken from the entire population, ensuring every member has an equal chance of being included.
### Why is an unbiased estimator preferred?
- [x] Because it ensures, on average, the estimates are correct.
- [ ] Because it always provides the best estimate.
- [ ] Because it minimizes variability.
- [ ] Because it adjusts estimates for sampling errors.
> **Explanation:** An unbiased estimator is preferred as it ensures that over multiple samples, the estimates average out to the true population parameter, leading to accurate inference about the population.
### What does the term 'expected value' refer to?
- [x] The mean value of the estimator's sampling distribution.
- [ ] The median value of the sample data.
- [ ] The mode of the population data.
- [ ] The range of the sampling distribution.
> **Explanation:** The expected value refers to the mean value of an estimator's sampling distribution, and for unbiased estimators, this expected value equals the population parameter.
### Are biased estimators never used in statistics?
- [ ] Yes, they are always inferior to unbiased estimators.
- [ ] Only when unbiased estimators are unavailable.
- [x] No, sometimes they are used if they offer other advantages.
- [ ] Only in theoretical contexts.
> **Explanation:** Biased estimators can be used in practice when they provide lower variance or are simpler to compute and interpret, thereby being more practical despite their bias.
### What is the main difference between a biased and an unbiased estimator?
- [x] Whether their expected value is equal to the population parameter or not.
- [ ] Whether they minimize the error or not.
- [ ] Whether they have high variability or not.
- [ ] Whether they adjust for sample size or not.
> **Explanation:** The main difference lies in the expected value; an unbiased estimator has an expected value equal to the population parameter, while a biased estimator does not.
### Can the mean of a random sample be an unbiased estimator of the population mean?
- [x] Yes, when the sample is randomly selected.
- [ ] No, only the median can be unbiased.
- [ ] Yes, only if the sample size is large.
- [ ] No, the mean is never unbiased.
> **Explanation:** The mean of a random sample is an unbiased estimator of the population mean if the sample is random, as the expected value of the sample mean equals the population mean.
### What is one method to verify if an estimator is unbiased?
- [ ] Comparing it with different estimators.
- [x] Calculating its expected value and comparing it to the parameter.
- [ ] Using a larger sample size.
- [ ] Ensuring identical samples are used.
> **Explanation:** To verify if an estimator is unbiased, calculate its expected value and compare it to the true population parameter. If they are equal, the estimator is unbiased.
### What role does sample size play in unbiased estimation?
- [ ] It ensures unbiasedness directly.
- [ ] It affects variability but not bias.
- [x] Larger sample sizes increase the reliability of the estimator.
- [ ] It guarantees unbiased estimates.
> **Explanation:** While sample size does not directly affect unbiasedness, larger sample sizes generally increase the reliability and precision of an estimator by reducing variability.
### What is the impact of biased estimators with lower variance compared to unbiased estimators with higher variance?
- [ ] They are always preferred due to lower bias.
- [x] They may be preferred in practical applications for stability.
- [ ] They are not preferred due to inherent bias.
- [ ] They are used only in theoretical studies.
> **Explanation:** Biased estimators with lower variance might be preferred in practical applications since they offer more stable and consistent estimates compared to unbiased estimators with higher variance.
Thank you for exploring the intricate details of unbiased estimators and testing your knowledge with our sample quiz questions. Continue your statistical journey with confidence!
