Which Of The Following Is A Biased Estimator

Let's dive into the concept of biased estimators in statistics, exploring what they are, why they occur, and how to identify them. In statistics, an estimator is a rule or formula used to estimate a population parameter based on sample data. A biased estimator is one that, on average, systematically overestimates or underestimates the true value of the parameter it is intended to estimate. Understanding bias in estimators is crucial for making accurate inferences and decisions based on data.

Understanding Estimators and Bias

Before delving into the specifics of biased estimators, let's clarify the fundamental concepts.

• Parameter: A parameter is a numerical value that describes a characteristic of a population. For example, the population mean ($\mu$) and the population variance ($\sigma^2$) are parameters.

• Estimator: An estimator is a statistic used to estimate a population parameter. It is a function of the sample data. For example, the sample mean ($\bar{x}$) is an estimator of the population mean ($\mu$), and the sample variance ($s^2$) is an estimator of the population variance ($\sigma^2$).

• Estimate: An estimate is the specific value obtained when an estimator is applied to a particular sample.

• Bias: Bias is the difference between the expected value of the estimator and the true value of the parameter being estimated. Mathematically, if $\hat{\theta}$ is an estimator of the parameter $\theta$, then the bias of $\hat{\theta}$ is given by:

  $Bias(\hat{\theta}) = E(\hat{\theta}) - \theta$

  If $Bias(\hat{\theta}) = 0$, then $\hat{\theta}$ is an unbiased estimator. If $Bias(\hat{\theta}) \neq 0$, then $\hat{\theta}$ is a biased estimator.

Common Examples of Biased Estimators

Several estimators are known to be biased under certain conditions. Here are a few common examples:

1. Sample Variance (Uncorrected):

   The sample variance, when calculated with a denominator of n (the sample size), is a biased estimator of the population variance. The formula for the uncorrected sample variance is:

   $s^2_{uncorrected} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

   This estimator tends to underestimate the population variance. The expected value of the uncorrected sample variance is:

   $E(s^2_{uncorrected}) = \frac{n-1}{n} \sigma^2$

   As you can see, $E(s^2_{uncorrected})$ is not equal to $\sigma^2$, indicating that the uncorrected sample variance is biased.

2. Maximum Likelihood Estimators (MLEs):

   While MLEs have many desirable properties (such as consistency and asymptotic efficiency), they can sometimes be biased, especially in small samples. Consistency means that as the sample size increases, the estimator converges to the true parameter value. However, for finite sample sizes, the MLE can be biased.

3. Ratio Estimators:

   Ratio estimators are used to estimate the ratio of two population parameters. These estimators are generally biased, although the bias tends to decrease as the sample size increases.

4. Selection Bias:

   Selection bias occurs when the sample is not representative of the population due to the method of selecting the sample. This can lead to biased estimates of population parameters. For example, if you survey only people who voluntarily respond to an online poll, the results may not accurately reflect the opinions of the entire population.

5. Omitted Variable Bias in Regression:

   In regression analysis, omitted variable bias occurs when a relevant variable is excluded from the regression model. This can cause the estimated coefficients of the included variables to be biased.

Why Does Bias Occur?

Bias in estimators can arise for various reasons, including:

• Mathematical Properties: Some estimators are inherently biased due to their mathematical formulation. The uncorrected sample variance falls into this category.

• Sampling Methods: The way a sample is selected can introduce bias. Non-random sampling methods, such as convenience sampling or voluntary response sampling, are particularly prone to bias.

• Model Misspecification: If the statistical model used to estimate parameters is incorrect or incomplete, the resulting estimators may be biased. Omitted variable bias in regression is an example of this.

• Small Sample Sizes: Some estimators that are asymptotically unbiased (i.e., unbiased as the sample size approaches infinity) may exhibit significant bias in small samples.

Identifying Biased Estimators

Identifying whether an estimator is biased involves both theoretical and empirical approaches.

1. Theoretical Analysis:

   • Calculate the Expected Value: The most direct way to determine if an estimator is biased is to calculate its expected value and compare it to the true value of the parameter. If the expected value differs from the parameter value, the estimator is biased.

   • Mathematical Derivation: Use mathematical techniques to derive the properties of the estimator and determine if it is biased. This often involves using probability theory and statistical inference.

2. Empirical Analysis:

   • Simulation Studies: Conduct simulation studies to evaluate the performance of the estimator under different conditions. Generate multiple random samples from a known population, calculate the estimator for each sample, and then compare the average of the estimates to the true parameter value. If the average estimate consistently differs from the true value, the estimator is likely biased.

   • Bootstrapping: Use bootstrapping techniques to estimate the bias of an estimator. Bootstrapping involves resampling from the observed data to create multiple simulated samples. The estimator is calculated for each bootstrap sample, and the bias is estimated as the difference between the average of the bootstrap estimates and the estimate calculated from the original sample.

Correcting for Bias

If an estimator is found to be biased, several methods can be used to correct for the bias.

1. Bias Correction Formulas:

   In some cases, it is possible to derive a bias correction formula that adjusts the estimator to remove or reduce the bias. For example, the corrected sample variance is:

   $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

   This estimator is unbiased for the population variance.

2. Jackknife Estimation:

   The jackknife is a resampling technique used to estimate the bias and variance of an estimator. It involves systematically leaving out one observation at a time from the sample, calculating the estimator for each subsample, and then using these estimates to estimate the bias.

3. Bootstrap Bias Correction:

   Bootstrapping can also be used to correct for bias. The bias is estimated using bootstrap samples, and then this estimate is used to adjust the original estimator.

4. Alternative Estimators:

   In some cases, it may be possible to find an alternative estimator that is less biased or unbiased. For example, in regression analysis, using a different model specification or estimation technique may reduce omitted variable bias.

Example: Biased Estimator - Uncorrected Sample Variance

Let's illustrate the concept of a biased estimator with an example. Suppose we have a population with a known variance $\sigma^2$. We want to estimate this variance using a sample of size n. We consider two estimators: the uncorrected sample variance ($s^2_{uncorrected}$) and the corrected sample variance ($s^2$).

1. Uncorrected Sample Variance:

   $s^2_{uncorrected} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}$

   As mentioned earlier, the expected value of this estimator is:

   $E(s^2_{uncorrected}) = \frac{n-1}{n} \sigma^2$

   Since $E(s^2_{uncorrected}) \neq \sigma^2$, the uncorrected sample variance is a biased estimator. It tends to underestimate the population variance.

2. Corrected Sample Variance:

   $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$

   The expected value of this estimator is:

   $E(s^2) = \sigma^2$

   Since $E(s^2) = \sigma^2$, the corrected sample variance is an unbiased estimator.

To demonstrate this with a simulation, let's generate random samples from a normal distribution with mean $\mu = 0$ and variance $\sigma^2 = 1$. We will calculate both the uncorrected and corrected sample variances for each sample and compare their averages to the true variance.

import numpy as np

# Parameters
mu = 0
sigma_sq = 1
n = 30  # Sample size
num_samples = 10000  # Number of samples

# Generate random samples
samples = np.random.normal(mu, np.sqrt(sigma_sq), size=(num_samples, n))

# Calculate sample means
sample_means = np.mean(samples, axis=1)

# Calculate uncorrected sample variances
uncorrected_variances = np.sum((samples - sample_means[:, np.newaxis])**2, axis=1) / n

# Calculate corrected sample variances
corrected_variances = np.sum((samples - sample_means[:, np.newaxis])**2, axis=1) / (n - 1)

# Calculate the average of the estimators
avg_uncorrected_variance = np.mean(uncorrected_variances)
avg_corrected_variance = np.mean(corrected_variances)

# Print the results
print(f"True variance: {sigma_sq}")
print(f"Average uncorrected variance: {avg_uncorrected_variance}")
print(f"Average corrected variance: {avg_corrected_variance}")

# Calculate the bias
bias_uncorrected = avg_uncorrected_variance - sigma_sq
bias_corrected = avg_corrected_variance - sigma_sq

print(f"Bias of uncorrected variance: {bias_uncorrected}")
print(f"Bias of corrected variance: {bias_corrected}")

The output of this simulation will show that the average uncorrected variance is less than the true variance, indicating a negative bias, while the average corrected variance is very close to the true variance, indicating a negligible bias.

Impact of Biased Estimators

The use of biased estimators can have significant consequences in statistical inference and decision-making.

• Inaccurate Inferences: Biased estimators can lead to inaccurate conclusions about population parameters. For example, if you underestimate the population variance, you may underestimate the uncertainty in your estimates and make overly optimistic predictions.

• Poor Decision-Making: Decisions based on biased estimates may be suboptimal or even incorrect. This can have serious implications in fields such as finance, healthcare, and engineering.

• Misleading Results: In scientific research, using biased estimators can lead to misleading results and incorrect interpretations of data. This can undermine the validity of research findings and hinder scientific progress.

Bias-Variance Tradeoff

In some situations, there is a tradeoff between bias and variance. It may be possible to reduce the bias of an estimator, but this often comes at the cost of increasing its variance, and vice versa. The mean squared error (MSE) is a common metric used to evaluate the overall performance of an estimator, taking into account both bias and variance:

$MSE(\hat{\theta}) = E[(\hat{\theta} - \theta)^2] = Var(\hat{\theta}) + [Bias(\hat{\theta})]^2$

The MSE represents the average squared difference between the estimator and the true parameter value. An estimator with a lower MSE is generally preferred, as it provides a more accurate estimate of the parameter.

Conclusion

Understanding biased estimators is crucial for anyone working with data. By recognizing the sources of bias and using appropriate techniques to identify and correct for bias, it is possible to obtain more accurate estimates of population parameters and make better decisions based on data. While some estimators are inherently biased due to their mathematical properties, others may be biased due to sampling methods or model misspecification. Careful consideration of these factors is essential for ensuring the validity and reliability of statistical analyses.