Biased & Unbiased Point Estimates

Unit: Sampling Distributions

Chapter: Biased & Unbiased Point Estimates

Reference: – Population & Sample, Point estimates & Parameters, Accuracy Bias, Unbiased point estimates, Interpreting & Comparing, Mean & Variance of sample means, Sample proportion & Bias, Maximum likelihood estimation, Methos of moments estimation, Sample size & estimation, Application & Examples.

After studying this chapter, you should be able to:

  • Point estimates, Parameters & Bias in Point Estimates.
  • Unbiased point estimates, Mean & Variance of Sample means.
  • Sample Proportion & Maximum livelihood estimation.
  • Method of Moments & Sample size estimation.

Point Estimates, Parameters & Bias in Point Estimates

Point Estimates:

  • A point estimate is a single value that is used to approximate an unknown population parameter based on sample data.
  • Point estimates provide a way to make educated guesses about population characteristics without having to observe the entire population.
  • Common point estimates include the sample mean, sample proportion, and sample variance.

Parameters:

4. Parameters are numerical characteristics of a population that we aim to estimate using sample data.

 

  • Examples of parameters include the population mean, population proportion, and population standard deviation.
  • Parameters are typically fixed and unknown, making them the focus of statistical inference.

Bias in Point Estimates:

7. Bias refers to the systematic tendency of a point estimate to consistently deviate from the true population parameter.

 

  • A point estimate is biased if, on average, it overestimates or underestimates the true parameter value.
  • Bias can arise from the sampling method, measurement errors, or other sources of systematic error.
  • Bias in point estimates can lead to inaccurate and misleading conclusions about the population.

Reducing Bias:

1. Unbiased point estimates are preferred because they, on average, provide accurate estimates of the population parameter.

  • Techniques like random sampling and proper study design can help reduce bias in point estimates.
  • Adjusting for bias involves using correction factors or more sophisticated statistical methods to obtain unbiased estimates.

Bias Correction Examples:

2. In estimating a population proportion, the sample proportion can be biased, but dividing by the correction factor (n-1) instead of n reduces bias in the estimate.

  • In estimating population variance, using the Bessel's correction (n-1) instead of n corrects the bias in the sample variance.

Unbiased Point Estimates, Mean & Variance of Sample Means

 

Unbiased Point Estimates:

  1. An unbiased point estimate is a statistic that, on average, accurately estimates the population parameter it represents.
  2. Unbiasedness implies that if we repeatedly take random samples from a population and calculate the point estimate each time, the average of these estimates will be equal to the true population parameter.
  3. Unbiased point estimates are preferred because they do not systematically overestimate or underestimate the population parameter.

 

Mean of Sample Means:

 

  1. The mean of sample means, often denoted as "x̄" (x-bar), is the average of all possible sample means of a given sample size that can be drawn from a population.
  2. The mean of sample means is also referred to as the "expected value of the sample mean."
  3. According to the Central Limit Theorem (CLT), when sample sizes are sufficiently large (usually n ≥ 30), the distribution of sample means becomes approximately normal, regardless of the underlying population distribution.

 

Variance of Sample Means:

 

  • The variance of sample means measures the spread or variability of the distribution of sample means around the population mean.
  • The variance of sample means is influenced by two factors: the population variance (σ²) and the sample size (n).
  • The formula for the variance of sample means is given by: Var(x̄) = σ² / n, where σ² is the population variance and n is the sample size.
  • As the sample size increases, the variance of sample means decreases, leading to a more precise estimate of the population mean.

Sample Proportion & Maximum Livelihood Estimation

Sample Proportion:

  • The sample proportion, denoted by "p̂" (p-hat), is a point estimate of the population proportion based on sample data.
  • It represents the proportion of successes (or events of interest) in the sample.
  • The sample proportion is used to estimate the population proportion, which is a parameter of interest.
  • The sample proportion is calculated as the ratio of the number of successes to the total sample size.

 

Maximum Likelihood Estimation (MLE):

  • Maximum Likelihood Estimation (MLE) is a method used to estimate the parameters of a statistical model based on observed data.
  • MLE aims to find the parameter values that maximize the likelihood function, which quantifies how well the model explains the observed data.
  • In the context of sample proportion, MLE seeks the value of the population proportion that makes the observed sample outcomes most probable.
  • MLE provides estimates that are efficient and asymptotically unbiased as the sample size increases.

Likelihood Function:

  • The likelihood function is a probability distribution function that represents the probability of observing the given sample data for different values of the parameter.
  • MLE involves finding the parameter value that maximizes the likelihood function, effectively making the observed data most likely under that parameter.

 

Applicability of MLE:

 

  • MLE is widely used in various fields, including biology, economics, engineering, and social sciences, to estimate unknown parameters.
  • MLE estimators are often preferred due to their desirable statistical properties, such as asymptotic efficiency.

Procedure for MLE:

  • To perform MLE, formulate the likelihood function based on the observed data and parameter of interest.
  • Take the derivative of the likelihood function with respect to the parameter and set it equal to zero to find the maximum.
  • Solve for the parameter value that maximizes the likelihood function to obtain the MLE estimate.

Method of Moments & Sample Size Estimation

Method of Moments:

  1. The Method of Moments (MoM) is a statistical technique used to estimate the parameters of a population distribution based on moments of the sample data.
  2. Moments are mathematical measures of the shape and location of a distribution, such as mean, variance, skewness, and kurtosis.
  3. MoM seeks to equate the sample moments (usually up to a certain order) with the corresponding population moments and solve for the parameter estimates.

 

Procedure for Method of Moments:

  1. Identify a suitable mathematical model or distribution that describes the data.
  2. Express the population moments (e.g., mean, variance) in terms of the distribution's parameters.
  3. Equate the sample moments (calculated from the data) with the corresponding population moments and solve for the parameter estimates.

Advantages of Method of Moments:

  1. MoM provides a simple and intuitive way to estimate population parameters.
  2. It can be used even when complex statistical distributions are involved, provided the moments exist and are well-defined.

 

Limitations of Method of Moments:

  • MoM may not always produce accurate estimates, especially for small sample sizes or when moments are poorly behaved.
  • It may not perform well for distributions with heavy tails or highly skewed data.

Sample Size Estimation:

  • Sample size estimation is the process of determining the number of observations needed in a sample to achieve a certain level of accuracy and confidence in statistical analysis.
  • Adequate sample size is crucial for obtaining reliable and meaningful results in statistical inference.

 

Factors Influencing Sample Size:

  Desired level of confidence (e.g., 95% confidence interval).

  •       Margin of error (precision) around the estimate.

Variability or expected standard deviation of the population.

 

Calculating Sample Size:

  • Sample size calculations often involve formulas based on the desired level of confidence, margin of error, and variability.
  • Software tools and statistical calculators are available to assist in sample size determination for various study designs and analyses.

 

 

Example: Estimating Average Income

 

Suppose you are conducting a survey to estimate the average income of households in a certain city. You randomly select a sample of 100 households and collect their income data. The true average income of all households in the city is $50,000.

Solution: – Biased Estimate:

 

Let's say that due to non-response bias, some higher-income households are less likely to participate in the survey. As a result, your sample tends to underrepresent high-income households. This leads to a biased estimate of the average income.

 

Suppose the average income in your sample is $48,000. This estimate is biased because it consistently underestimates the true average income due to the non-response bias.

 

Unbiased Estimate:

 

To correct for the bias, you can use a weighted approach. You know that the sample is biased towards lower incomes, so you can assign higher weights to the incomes of the higher-income households that did participate. This will help in adjusting the estimate to be closer to the true population average.

 

Suppose you calculate the weighted average income to be $49,000. This estimate is closer to the true average of $50,000 and is an unbiased estimate because, on average, it accurately represents the population parameter.

 

Key Points

  • Definition: Point estimates are single values calculated from sample data to estimate unknown population parameters.
  • Bias: Bias refers to a consistent tendency of a point estimate to systematically overestimate or underestimate the true population parameter.
  • Biased Estimate: A biased estimate consistently deviates from the true parameter value in the same direction due to flaws in the sampling or measurement process.
  • Unbiased Estimate: An unbiased estimate, on average, equals the true parameter value when considering all possible random samples.
  • Selection Bias: Occurs when the method of selecting the sample systematically favors certain groups or characteristics, leading to an inaccurate estimate.
  • Non-Response Bias: Arises when non-response by certain individuals or groups in a sample affects the estimate of the population parameter.
  • Measurement Bias: Results from errors or inaccuracies in the measurement instrument or process used to collect data.
  • Mitigating Bias:

 

  • Weighted Estimates: Adjusting the contribution of each data point based on its perceived reliability can reduce bias in estimates.
  • Random Sampling: Employing random sampling techniques can help mitigate bias by ensuring all individuals or groups have an equal chance of being included.
  • Stratified Sampling: Dividing the population into subgroups and then randomly sampling from each subgroup can address bias and improve estimates.
  • Unbiased Estimators:

 

  • Definition: An estimator is unbiased if, on average, its expected value is equal to the true population parameter it aims to estimate.
  • Sample Mean and Proportion: The sample mean and sample proportion are often unbiased estimators when calculated from random samples.
  • Bias vs. Variability:

 

  • Trade-off: While unbiasedness is desirable, it's important to balance bias and variability; reducing bias may increase variability.
  • Efficiency: Biased estimators can sometimes have lower variability (greater precision) than unbiased estimators but may still lead to inaccurate results.
  • Real-world Application:

 

  • Election Polling Example: In election polling, biased samples can lead to inaccurate predictions, while unbiased samples are more likely to reflect the true voting patterns of the population.

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