Which Of The Following Is A Biased Estimator

Let's delve into the concept of biased estimators and identify which ones, among a given set, fall into this category. In statistics, an estimator is a rule or formula used to estimate a population parameter based on a sample of data. A biased estimator is one that, on average, overestimates or underestimates the true value of the parameter it's trying to estimate. Understanding bias is crucial for making sound inferences and decisions based on statistical analysis.

Understanding Bias in Estimation

Before pinpointing specific biased estimators, let's solidify our grasp of what statistical bias truly means.

What is Bias?

Bias, in the context of estimation, refers to a systematic difference between the expected value of an estimator and the true value of the parameter being estimated. Imagine repeatedly drawing samples from a population and using the same estimator each time. If the average of all the estimates obtained is not equal to the true population parameter, then the estimator is biased.

Mathematically, if we let θ represent the true population parameter and θ̂ represent an estimator of θ, then the bias of the estimator is defined as:

Bias( θ̂ ) = E( θ̂ ) - θ

Where E( θ̂ ) is the expected value (or average) of the estimator θ̂.

• If Bias( θ̂ ) > 0, the estimator overestimates the true parameter on average.
• If Bias( θ̂ ) < 0, the estimator underestimates the true parameter on average.
• If Bias( θ̂ ) = 0, the estimator is unbiased. This means that on average, the estimator hits the true value.

Why Does Bias Matter?

Bias can lead to incorrect conclusions and poor decisions. If an estimator consistently overestimates a parameter, you might be led to believe that a quantity is larger than it actually is. Conversely, underestimation can lead to a failure to recognize important effects or trends. While unbiasedness is a desirable property, it's important to remember that it's not the only consideration. We also care about the variance of the estimator, which measures how much the estimates vary from sample to sample.

Bias-Variance Tradeoff:

Sometimes, we might choose a slightly biased estimator if it has a significantly lower variance than an unbiased estimator. This is known as the bias-variance tradeoff. A lower variance means that the estimates are more consistent, even if they are slightly off on average. The best estimator minimizes the mean squared error (MSE), which combines both bias and variance:

MSE( θ̂ ) = Variance( θ̂ ) + Bias( θ̂ )²

Common Examples of Biased Estimators

Now, let's look at some specific examples of estimators that are known to be biased, along with the reasons why:

1. Sample Variance (when calculated with n as the denominator):

   • The Estimator: s² = Σ(x_i - x̄)² / n, where x_i are the individual data points, x̄ is the sample mean, and n is the sample size.
   • Why it's Biased: This estimator underestimates the true population variance. The reason is that when we calculate the sample variance, we are using the sample mean (x̄) as an estimate of the population mean (μ). However, the sample mean is calculated from the same data as the variance. This forces the data points to be, on average, closer to the sample mean than they would be to the true population mean. Consequently, the sum of squared deviations from the sample mean is smaller than the sum of squared deviations from the population mean would be, leading to an underestimation of the variance.
   • The Correction: To correct for this bias, we use n-1 (degrees of freedom) in the denominator instead of n. The corrected sample variance is: s² = Σ(x_i - x̄)² / (n-1). This provides an unbiased estimator of the population variance. The term n-1 is used because one degree of freedom is "lost" when estimating the mean from the same sample data.
   • Practical Implication: If you use the biased version of the sample variance, your confidence intervals will be narrower than they should be, and you might be more likely to reject a true null hypothesis (Type I error).

2. Maximum Likelihood Estimator (MLE) for Variance of a Normal Distribution:

   • The Estimator: The MLE for the variance of a normal distribution is similar to the biased sample variance: σ̂² = Σ(x_i - μ̂)² / n, where μ̂ is the MLE for the mean (which is the sample mean).
   • Why it's Biased: For the same reason as the biased sample variance (using the sample mean to estimate the population mean), the MLE for the variance of a normal distribution is also biased downwards.
   • Important Note: While MLEs are often asymptotically unbiased (meaning the bias decreases as the sample size increases), they can be biased for small sample sizes. MLEs possess many desirable properties, but unbiasedness is not guaranteed.

3. Ratio Estimators:

   • The Estimator: A ratio estimator is used to estimate the ratio of two population means (e.g., total sales per employee). It's calculated as the ratio of the sample means: (Σy_i / n) / (Σx_i / n) = Σy_i / Σx_i, where y_i and x_i are observations of two different variables.
   • Why it's Biased: Ratio estimators are generally biased unless the relationship between the numerator and denominator is linear and passes through the origin. The bias arises because the expected value of a ratio is not generally equal to the ratio of the expected values: E(Y/X) ≠ E(Y) / E(X). Jensen's inequality implies that E(g(X)) ≠ g(E(X)) for nonlinear functions g. The division operation is a non-linear operation, thus causing bias.
   • Example: Estimating the average yield of a crop per acre. If you have sample data on total yield and total acreage, dividing the total sample yield by the total sample acreage gives you a ratio estimate of the average yield per acre. This estimate will be biased unless the relationship between yield and acreage is perfectly linear and passes through the origin (which is unlikely).

4. Estimators in the Presence of Selection Bias:

   • The Scenario: Selection bias occurs when the sample is not representative of the population due to the way it was selected.
   • Why it's Biased: If the selection process favors certain individuals or groups, the resulting estimates will be biased towards the characteristics of the over-represented groups and away from the characteristics of the under-represented groups.
   • Example: A survey about customer satisfaction that only collects responses from customers who voluntarily provide feedback (e.g., through an online form). Customers who are very satisfied or very dissatisfied are more likely to respond than those who are moderately satisfied. This will bias the results, making the average satisfaction score either higher or lower than the true average satisfaction of all customers.

5. Regression to the Mean:

   • The Scenario: Regression to the mean is a statistical phenomenon where extreme values tend to be followed by values that are closer to the average.
   • Why it leads to Biased Interpretation: If you select a sample based on extreme values, and then measure the same variable again, you'll likely observe that the values have moved closer to the mean. It's tempting to attribute this change to some intervention or cause, but it's often simply due to regression to the mean. Failing to account for this can lead to biased interpretations of the data.
   • Example: Identifying students who score very low on a test and then providing them with tutoring. On a subsequent test, their scores will likely improve, even if the tutoring had no effect. This is because their initial low scores were likely partly due to random chance, and on the second test, their scores will regress towards their true average ability.

6. Omitted Variable Bias in Regression:

   • The Scenario: Omitted variable bias occurs when a relevant variable is not included in a regression model.
   • Why it's Biased: If the omitted variable is correlated with both the dependent variable and one or more of the included independent variables, the coefficients of the included variables will be biased. The estimated effect of the included variables will incorrectly reflect some of the effect of the omitted variable.
   • Example: Trying to estimate the effect of education on income, but failing to include a measure of ability. If ability is correlated with both education and income, the estimated effect of education on income will be biased upwards, because it will also be capturing some of the effect of ability.

7. Survivorship Bias:

   • The Scenario: This occurs when you only consider entities that have "survived" some process, overlooking those that did not. This can lead to a skewed understanding of the factors that contribute to success or failure.
   • Why it's Biased: By only looking at the survivors, you miss critical information about the entire population, leading to biased conclusions.
   • Example: Analyzing the success stories of startups without considering the many more that failed. This will give a distorted picture of the strategies and characteristics that lead to startup success.

Addressing Bias

While it's not always possible to eliminate bias completely, there are several strategies that can be used to mitigate its effects:

• Use Unbiased Estimators: Whenever possible, choose estimators that are known to be unbiased. For example, use the sample variance with n-1 in the denominator instead of n.
• Increase Sample Size: In many cases, the bias of an estimator decreases as the sample size increases. Larger samples provide more information about the population and reduce the impact of random fluctuations.
• Random Sampling: Use random sampling techniques to ensure that the sample is representative of the population. This helps to minimize selection bias.
• Control for Confounding Variables: In regression analysis, include all relevant variables in the model to control for confounding effects and reduce omitted variable bias.
• Awareness and Critical Thinking: Be aware of the potential sources of bias in your data and analysis. Critically evaluate your assumptions and methods to identify and address any biases that may be present.
• Cross-Validation: In machine learning, cross-validation techniques can help to assess the performance of a model on unseen data and detect potential biases.
• Sensitivity Analysis: Perform sensitivity analysis to assess how the results of your analysis change when you vary the assumptions or parameters. This can help to identify whether your conclusions are sensitive to certain biases.

FAQ about Biased Estimators

• Q: Is an unbiased estimator always better than a biased estimator?

  • A: Not necessarily. While unbiasedness is a desirable property, it's important to consider the variance of the estimator as well. A slightly biased estimator with a low variance might be preferable to an unbiased estimator with a high variance, especially if the mean squared error (MSE) is lower.

• Q: How can I tell if an estimator is biased?

  • A: The best way to determine if an estimator is biased is to derive its expected value mathematically. If the expected value is not equal to the true value of the parameter being estimated, then the estimator is biased. Simulation studies can also be used to estimate the bias of an estimator.

• Q: Can bias be eliminated completely?

  • A: In some cases, bias can be eliminated by using unbiased estimators or by correcting for bias in biased estimators. However, in other cases, it may not be possible to eliminate bias completely. It's important to be aware of the potential sources of bias and to take steps to mitigate their effects.

• Q: What's the difference between bias and variance?

  • A: Bias refers to the systematic difference between the expected value of an estimator and the true value of the parameter being estimated. Variance, on the other hand, measures the variability of the estimator around its expected value. A high variance means that the estimates are more spread out, while a low variance means that the estimates are more consistent.

• Q: Are all maximum likelihood estimators (MLEs) unbiased?

  • A: No, MLEs are not always unbiased. While MLEs have many desirable properties, such as consistency and asymptotic efficiency, they can be biased, especially for small sample sizes.

Conclusion

Identifying biased estimators is a critical step in conducting sound statistical analysis. Understanding the sources of bias and the strategies for mitigating its effects allows us to draw more accurate conclusions and make better decisions based on data. While unbiasedness is a desirable property, it should be considered in conjunction with other factors, such as variance and mean squared error, to choose the best estimator for a particular problem. Recognizing the potential for bias and taking steps to address it are essential for responsible and effective statistical practice. By carefully considering the properties of different estimators and the potential sources of bias, we can improve the quality of our statistical inferences and make more informed decisions.