Which Of The Following Is A Biased Estimator

Let's delve into the core concept of biased estimators in statistics, dissecting what they are, why they matter, and how to identify them. Understanding bias is critical for anyone working with data, from students to seasoned professionals.

Understanding Estimators and Their Properties

In statistics, an estimator is a rule or a formula, typically expressed as a function, that is used to estimate a population parameter based on a sample of data. Population parameters are characteristics of the entire group we are interested in, such as the population mean or population variance. However, it's often impractical or impossible to measure these parameters directly. Instead, we take a sample from the population and use an estimator to calculate an estimate of the parameter.

Several desirable properties characterize a "good" estimator:

• Unbiasedness: An estimator is unbiased if its expected value equals the true population parameter. In other words, if we were to repeatedly draw samples from the population and calculate the estimate using our estimator, the average of these estimates would converge to the true population parameter.
• Efficiency: An estimator is efficient if it has a small variance. This means that the estimates obtained from different samples will be clustered closely together, providing a more precise estimate of the population parameter.
• Consistency: An estimator is consistent if it converges to the true population parameter as the sample size increases. This means that with larger samples, we can be more confident that our estimate is close to the true value.
• Sufficiency: An estimator is sufficient if it uses all the information in the sample that is relevant to estimating the parameter. A sufficient estimator captures all the information about the parameter contained in the data.

What is a Biased Estimator?

A biased estimator is an estimator whose expected value does not equal the true value of the population parameter being estimated. This means that, on average, the estimator will systematically overestimate or underestimate the parameter. The bias is the difference between the expected value of the estimator and the true parameter value.

Bias = E[θ̂] - θ

Where:

• E[θ̂] is the expected value of the estimator θ̂ (theta-hat)
• θ is the true value of the parameter

Consequences of Using a Biased Estimator:

Using a biased estimator can lead to several problems:

• Inaccurate conclusions: Biased estimates can lead to incorrect conclusions about the population being studied. This can have serious consequences in fields such as medicine, economics, and engineering, where decisions are often based on statistical analyses.
• Poor predictions: If a model is built using biased estimates, it may make poor predictions about future observations.
• Misleading inferences: Biased estimates can lead to misleading inferences about the relationships between variables.

Common Sources of Bias in Estimators

Bias can arise from various sources. Understanding these sources is crucial for identifying and mitigating bias in statistical analysis. Here are some common culprits:

1. Selection Bias: This occurs when the sample selected is not representative of the population.
   • Example: Polling only people who own smartphones to estimate the average income of a population would introduce selection bias, as it excludes people with lower incomes who may not own smartphones.
2. Omitted Variable Bias: This arises when a relevant variable is excluded from a statistical model. If the omitted variable is correlated with both the independent and dependent variables, it can bias the estimated effect of the included variables.
   • Example: Estimating the effect of education on income without controlling for parental wealth can lead to omitted variable bias, as parental wealth is likely correlated with both education and income.
3. Measurement Error: This occurs when the data collected is inaccurate or unreliable.
   • Example: Using a faulty scale to measure weights in a study would introduce measurement error, leading to biased estimates of population weight.
4. Sampling Bias: This is similar to selection bias but specifically relates to the method used to select the sample.
   • Example: Using a convenience sample (e.g., surveying people in a shopping mall) may not be representative of the entire population.
5. Publication Bias: This occurs when research with statistically significant results is more likely to be published than research with null results. This can lead to an overestimation of the true effect size in meta-analyses and literature reviews.
6. Response Bias: This arises when respondents provide inaccurate or untruthful answers to survey questions.
   • Example: People may underreport their alcohol consumption or overreport their charitable donations due to social desirability bias.
7. Survivorship Bias: This occurs when only the "surviving" members of a group are considered, leading to a distorted view of the group's overall performance.
   • Example: Studying only successful companies without considering the many that failed can lead to an overestimation of the factors contributing to success.

Examples of Biased Estimators

Let's explore some specific examples of biased estimators that frequently appear in statistical contexts.

1. Sample Variance (without Bessel's Correction):

   The sample variance, calculated without Bessel's correction (n-1 in the denominator), is a biased estimator of the population variance. The formula for the sample variance is:

   s² = Σ(x_i - x̄)² / n

   Where:

   • x_i is each individual data point in the sample
   • x̄ is the sample mean
   • n is the sample size

   This estimator is biased because it tends to underestimate the population variance. This is because it uses the sample mean (x̄) to calculate the deviations, which is already centered around the sample data. Therefore, the deviations tend to be smaller than if we were using the true population mean (μ), which is unknown.

   The expected value of this estimator is:

   E[s²] = ((n-1)/n) σ²

   Where σ² is the true population variance. As you can see, E[s²] is not equal to σ², demonstrating the bias.

   Bessel's Correction: To correct for this bias, we use Bessel's correction, which involves dividing by (n-1) instead of n:

   s² = Σ(x_i - x̄)² / (n-1)

   This provides an unbiased estimator of the population variance.

2. Maximum Likelihood Estimator (MLE) for Variance in Certain Cases:

   While MLEs are often asymptotically unbiased (approaching unbiasedness as the sample size increases), they can be biased for finite sample sizes. For instance, in estimating the variance of a normal distribution, the MLE can be biased similar to the uncorrected sample variance mentioned above.

3. Ratio Estimator:

   A ratio estimator is used to estimate the ratio of two population means. It is calculated as the ratio of the sample means of two variables. While simple to compute, it's generally a biased estimator, especially for small sample sizes or when the relationship between the two variables is not linear. The bias decreases as the sample size increases.

4. Estimator in Omitted Variable Bias (Regression):

   As discussed earlier, in multiple regression, if a relevant variable is omitted from the model and is correlated with both the included independent variables and the dependent variable, the estimators of the coefficients of the included variables will be biased.

5. Self-Selection Bias in Surveys:

   Consider a survey where participation is voluntary. Individuals who have strong opinions or experiences related to the survey topic are more likely to participate. This can lead to a biased representation of the population's views. For example, online reviews often suffer from self-selection bias, as people are more likely to leave a review if they had a particularly positive or negative experience.

Identifying Biased Estimators

Identifying biased estimators can be challenging, but several strategies can help:

1. Mathematical Derivation: The most rigorous way is to mathematically derive the expected value of the estimator and compare it to the true population parameter. If they are not equal, the estimator is biased. This often involves using statistical theory and calculus.

2. Simulations: Simulations can be a practical way to assess bias, especially when a mathematical derivation is difficult. This involves:

   • Generating many random samples from a population with known parameters.
   • Calculating the estimate using the estimator for each sample.
   • Averaging the estimates across all samples.
   • Comparing the average estimate to the true population parameter.

   If the average estimate differs significantly from the true parameter, the estimator is likely biased.

3. Understanding the Properties of the Estimator: Being familiar with the theoretical properties of different estimators is crucial. For example, knowing that the uncorrected sample variance is biased, or that ratio estimators are prone to bias, allows you to choose alternative, less biased estimators when possible.

4. Domain Expertise and Critical Thinking: Consider the context of the problem and potential sources of bias. Does the sampling method introduce selection bias? Are there any omitted variables that could be affecting the results? Critical thinking and domain expertise are essential for identifying potential sources of bias.

5. Sensitivity Analysis: Perform a sensitivity analysis by varying assumptions and inputs to the estimation process. This helps understand how sensitive the results are to changes in these factors, which can reveal potential sources of bias.

Correcting for Bias

While it is always preferable to use an unbiased estimator from the start, sometimes we are stuck with a biased one. There are several techniques to correct for bias:

1. Bessel's Correction: As demonstrated earlier, Bessel's correction corrects the bias in the sample variance estimator.

2. Jackknife and Bootstrap Resampling: These are resampling techniques that can be used to estimate and reduce bias in estimators. They involve creating multiple "pseudo-samples" from the original sample and using them to estimate the bias.

3. Bias Reduction Techniques in Regression: Techniques like adding control variables to address omitted variable bias, or using instrumental variables to address endogeneity, can reduce bias in regression models.

4. Weighting: In survey data, weighting can be used to adjust for unequal probabilities of selection and reduce selection bias. For example, if certain demographic groups are underrepresented in the sample, their responses can be weighted more heavily.

5. Calibration: This involves adjusting the estimates to match known population totals or other external benchmarks. This can help reduce bias by ensuring that the estimates are consistent with other reliable sources of information.

The Bias-Variance Tradeoff

It's important to understand the bias-variance tradeoff. In some situations, reducing bias can increase variance, and vice versa. For example, using a very complex model with many parameters can reduce bias but may lead to high variance, meaning the model is very sensitive to the specific data used for training and may not generalize well to new data. Conversely, using a very simple model can reduce variance but may lead to high bias, meaning the model is too simplistic and cannot capture the true relationships in the data. The goal is to find a balance between bias and variance that minimizes the overall error.

Conclusion

Understanding the concept of biased estimators is paramount in statistical analysis. Recognizing the potential sources of bias, identifying biased estimators, and employing techniques to correct for bias are crucial steps in ensuring accurate and reliable results. While eliminating bias is always desirable, the bias-variance tradeoff highlights the importance of finding the right balance to minimize overall error and draw meaningful conclusions from data. Always critically evaluate the estimators used in your analysis and consider the potential impact of bias on your findings.