Determine Which Of The Following Show Three Biased Estimators

Determining Biased Estimators: A Deep Dive into Statistical Inference

Understanding bias in estimators is crucial for accurate statistical inference. A biased estimator consistently overestimates or underestimates the true population parameter it's trying to estimate. This article will delve into the concept of biased estimators, providing a clear methodology to identify them and exploring examples to solidify understanding. We'll analyze several scenarios to determine which exhibit biased estimators, emphasizing the importance of unbiased estimation in statistical analysis.

What is an Estimator and Bias?

Before identifying biased estimators, let's define key terms:

• Estimator: An estimator is a statistic calculated from sample data that is used to estimate an unknown population parameter. For example, the sample mean (x̄) is an estimator for the population mean (μ).

• Bias: The bias of an estimator is the difference between its expected value (the average value of the estimator over all possible samples) and the true value of the population parameter. A biased estimator systematically deviates from the true value. Formally, if θ is the population parameter and θ̂ is its estimator, the bias is E[θ̂] - θ. If this difference is zero, the estimator is unbiased.

• Unbiased Estimator: An unbiased estimator has an expected value equal to the true population parameter. This means that, on average, the estimator will hit the target.

Identifying Biased Estimators: A Step-by-Step Approach

To determine if an estimator is biased, we need to follow these steps:

1. Identify the Population Parameter: Clearly define the population parameter you are trying to estimate. This is the true value you want to approximate.

2. Define the Estimator: Specify the statistic you're using to estimate the parameter. This is often a function of the sample data.

3. Find the Expected Value of the Estimator: Calculate the expected value of the estimator. This requires understanding the probability distribution of the sample data and applying relevant expectation rules. This step often involves complex mathematical calculations.

4. Compare the Expected Value to the True Parameter: Compare the expected value of the estimator calculated in step 3 to the true population parameter identified in step 1. If they are equal, the estimator is unbiased; otherwise, it is biased. The difference represents the bias.

5. Analyze the Bias: If the estimator is biased, analyze the direction and magnitude of the bias. Does it consistently overestimate or underestimate the true parameter? How large is the discrepancy? This helps to understand the implications of the bias.

Examples: Determining Biased Estimators

Let's consider several scenarios to illustrate how to identify biased estimators. We'll focus on three potential estimators and analyse their bias.

Scenario 1: Estimating Population Mean with the Sample Maximum

Let's say we want to estimate the population mean (μ) of a data set. We could consider several estimators:

• Estimator 1: Sample Mean (x̄) – This is the average of all the values in the sample. It's a well-known unbiased estimator for the population mean.

• Estimator 2: Sample Median (M) – The median is the middle value in a sorted sample. While often robust to outliers, it's generally a biased estimator of the population mean, except in symmetrical distributions.

• Estimator 3: Sample Maximum (Max) – This is the largest value in the sample. It is a severely biased estimator of the population mean, consistently overestimating it. The expected value of the sample maximum is heavily influenced by the upper tail of the distribution and will almost always be greater than the true population mean.

Analysis:

• Estimator 1 (x̄): The expected value of the sample mean is E[x̄] = μ. Therefore, the sample mean is an unbiased estimator of the population mean.

• Estimator 2 (M): The expected value of the sample median depends on the population distribution. For symmetrical distributions, E[M] ≈ μ, but for skewed distributions, it is biased.

• Estimator 3 (Max): The expected value of the sample maximum is always greater than μ. Therefore, the sample maximum is a highly biased estimator of the population mean.

Scenario 2: Estimating Population Variance

Consider estimating the population variance (σ²) using different estimators:

• Estimator 1: Sample Variance (s²) – This is the average squared deviation from the sample mean, calculated as Σ(xi - x̄)² / (n-1). This is an unbiased estimator of the population variance.

• Estimator 2: Biased Sample Variance (s̄²) – Similar to the sample variance, but calculated as Σ(xi - x̄)² / n. This estimator is biased, consistently underestimating the population variance.

Analysis:

• Estimator 1 (s²): The expected value of the sample variance (using the Bessel's correction) is E[s²] = σ². This is an unbiased estimator.

• Estimator 2 (s̄²): The expected value of the biased sample variance is E[s̄²] = [(n-1)/n]σ². This is a biased estimator; it underestimates the population variance.

Scenario 3: Estimating Population Proportion

Suppose we are estimating the population proportion (p) of individuals with a certain characteristic.

• Estimator 1: Sample Proportion (p̂) – This is the proportion of individuals with the characteristic in the sample. It is an unbiased estimator of the population proportion.

Analysis:

• Estimator 1 (p̂): The expected value of the sample proportion is E[p̂] = p. This is an unbiased estimator.

Implications of Biased Estimators

Using biased estimators can lead to inaccurate conclusions and flawed statistical inferences. The extent of the problem depends on the magnitude of the bias and the sample size. A small bias with a large sample may not be severely problematic, but a large bias can lead to significant errors in estimation.

Mitigating Bias

Several techniques can help mitigate bias:

• Using unbiased estimators: Choosing estimators known to be unbiased is the most straightforward approach.

• Bias correction: Some biased estimators can be corrected to reduce bias. For example, the Bessel's correction in calculating sample variance.

• Larger sample sizes: Increasing the sample size can reduce the impact of bias, although it doesn't eliminate it entirely.

• Bootstrapping: Resampling techniques like bootstrapping can provide estimates of bias and uncertainty, allowing for better understanding of estimator performance.

• Robust estimators: Using robust estimators, less sensitive to outliers, can minimize the influence of extreme values that might contribute to bias.

Conclusion

Determining whether an estimator is biased is crucial for reliable statistical inference. By following a systematic approach – defining the parameter, specifying the estimator, finding its expected value, and comparing it to the true parameter – we can identify biased estimators and understand their implications. While some bias might be acceptable in certain contexts, choosing unbiased estimators whenever possible is essential for generating accurate and trustworthy statistical results. Remember, the goal is to obtain estimates that are as close as possible to the true population parameters, and using unbiased estimators significantly contributes to this goal. Understanding the nuances of bias is a cornerstone of advanced statistical analysis.