Which Of The Following Is Considered An Estimator

In statistics, the concept of an estimator is fundamental to making inferences about population parameters based on sample data. An estimator is essentially a rule, typically expressed as a mathematical function, that tells you how to calculate an estimate of a population parameter from the data you have collected. Understanding what constitutes an estimator, its properties, and different types of estimators is crucial for anyone involved in data analysis, research, or decision-making based on statistical evidence.

What is an Estimator?

An estimator is a statistic used to infer the value of an unknown population parameter. It's important to distinguish between an estimator and an estimate.

Estimator: A function or a rule (formula) that describes how to calculate an estimate. It is a random variable because its value depends on the random sample selected.
Estimate: The specific value obtained when the estimator is applied to a particular sample of data. It is a fixed number.

For example, the sample mean (calculated from a sample of data) is an estimator of the population mean. If you calculate the sample mean from a specific dataset and find it to be 50, then 50 is the estimate of the population mean.

Key Properties of a Good Estimator

Not all estimators are created equal. Some estimators are better than others in terms of their ability to accurately and consistently estimate the population parameter. The key properties that define a "good" estimator include:

Unbiasedness: An estimator is unbiased if its expected value is equal to the true value of the population parameter. In other words, on average, the estimator will give you the correct answer. Mathematically, if θ is the true parameter and θ̂ is the estimator, then E(θ̂) = θ for an unbiased estimator.
Efficiency: An estimator is efficient if it has a small variance compared to other estimators. This means that the estimates obtained from the estimator will be more consistent and less spread out around the true parameter value. Efficiency is often measured relative to other estimators; the estimator with the smallest variance is considered the most efficient.
Consistency: An estimator is consistent if it converges to the true value of the population parameter as the sample size increases. Formally, as n (sample size) approaches infinity, the probability that the estimator θ̂ is close to the true parameter θ approaches 1. This ensures that with enough data, the estimator will provide a reliable estimate.
Sufficiency: A sufficient estimator is one that uses all the information in the sample that is relevant to estimating the parameter. In other words, no other estimator calculated from the same sample can provide additional information about the parameter being estimated.

Common Estimators and Their Applications

Several common estimators are used in statistical analysis, each suited for estimating different types of population parameters:

Sample Mean (x̄): This is the most common estimator for the population mean (μ). It is calculated by summing all the values in the sample and dividing by the sample size (n):

x̄ = (Σxᵢ) / n

The sample mean is an unbiased estimator of the population mean if the sample is randomly selected.
Sample Variance (s²): This is used to estimate the population variance (σ²), which measures the spread or dispersion of the data around the mean. The sample variance is calculated as:

s² = Σ(xᵢ - x̄)² / (n - 1)

Note that we divide by (n-1) instead of n to make the sample variance an unbiased estimator of the population variance. Dividing by n would result in a biased (under) estimate. The (n-1) term represents the degrees of freedom.
Sample Standard Deviation (s): This estimates the population standard deviation (σ), which is the square root of the variance.

s = √s²

The sample standard deviation is not an unbiased estimator of the population standard deviation. However, the bias decreases as the sample size increases.
Sample Proportion (p̂): This is used to estimate the population proportion (p), which represents the fraction of a population that possesses a certain characteristic. It is calculated as:

p̂ = x / n

where x is the number of observations in the sample with the characteristic and n is the sample size. The sample proportion is an unbiased estimator of the population proportion.
Least Squares Estimator: Used in regression analysis to estimate the coefficients of a linear model. This estimator minimizes the sum of the squared differences between the observed values and the values predicted by the model.
Maximum Likelihood Estimator (MLE): A very powerful and versatile estimation method. The MLE chooses the value of the parameter that maximizes the likelihood function, which represents the probability of observing the given data under different parameter values. MLEs often have desirable properties like consistency and asymptotic efficiency.

Examples of Identifying Estimators

Let's look at some scenarios and determine what would be considered an estimator:

Scenario 1: You want to estimate the average height of all students at a university. You randomly select 100 students and measure their heights.

Estimator: The sample mean of the 100 heights you measured. This is the rule you're using to estimate the population mean height. The formula x̄ = (Σxᵢ) / n is the estimator.
Estimate: The actual numerical value you calculate when you apply the sample mean formula to your data (e.g., 170 cm).

Scenario 2: You want to estimate the proportion of voters in a city who support a particular candidate. You conduct a survey of 500 randomly selected voters.

Estimator: The sample proportion of voters in your survey who support the candidate. This is the rule you're using to estimate the population proportion. The formula p̂ = x / n is the estimator.
Estimate: The specific percentage you calculate from your survey data (e.g., 55%).

Scenario 3: You want to determine the relationship between years of education and income. You collect data on both variables for a sample of adults and perform a linear regression.

Estimator: The least squares estimator used to calculate the coefficients (slope and intercept) of the regression line. The formula used to derive these coefficients is the estimator.
Estimate: The specific numerical values of the slope and intercept that result from applying the least squares method to your data.

Examples of what are NOT estimators:

The population parameter itself: The population mean (μ) is a parameter, not an estimator. An estimator is something you calculate from the sample to estimate the population parameter.
A single data point: A single observation from your sample is data, not an estimator. Estimators are functions of the entire sample data.
A pre-determined constant: If you decide to estimate the population mean by simply guessing the number 100, then 100 is just a constant, not an estimator. An estimator must be based on the sample data.
A theoretical distribution: The normal distribution is a probability distribution, not an estimator. You might use the normal distribution in conjunction with an estimator (e.g., constructing a confidence interval), but the distribution itself is not the estimator.

Biased vs. Unbiased Estimators: A Deeper Dive

Unbiasedness is a highly desirable property of an estimator. A biased estimator systematically overestimates or underestimates the population parameter. While unbiasedness is ideal, it's not always the most important factor in choosing an estimator. Sometimes, a slightly biased estimator with a much smaller variance (i.e., higher precision) is preferable to an unbiased estimator with a large variance.

Examples of Bias:

Sample Variance (using n instead of n-1): As mentioned earlier, if you calculate the sample variance by dividing by n instead of n-1, you will get a biased (under) estimate of the population variance. This is because dividing by n doesn't fully account for the fact that the sample mean is used to estimate the population mean, which introduces a degree of dependence among the data points.
Selection Bias: If your sample is not randomly selected from the population, your estimators are likely to be biased. For example, if you only survey people who are willing to respond to your survey, your results may not be representative of the entire population.
Non-response Bias: Similar to selection bias, non-response bias occurs when certain groups of people are less likely to respond to your survey than others. This can lead to biased estimates if the non-respondents have different characteristics than the respondents.

Why Use a Biased Estimator?

In some cases, it may be advantageous to use a biased estimator if it offers other desirable properties, such as lower mean squared error (MSE). MSE combines both the bias and the variance of an estimator into a single measure of its overall accuracy:

MSE(θ̂) = Variance(θ̂) + [Bias(θ̂)]²

Sometimes, reducing the variance of an estimator can lead to a lower MSE, even if it introduces a small amount of bias. This is particularly true when dealing with small sample sizes or complex models.

The Importance of Sample Size

The sample size plays a crucial role in the performance of an estimator. In general, larger sample sizes lead to more accurate and reliable estimates. This is because:

Reduced Variance: The variance of most estimators decreases as the sample size increases. This means that the estimates will be more tightly clustered around the true parameter value.
Increased Consistency: Consistent estimators converge to the true parameter value as the sample size approaches infinity.
More Representative Sample: Larger samples are more likely to be representative of the population, reducing the risk of selection bias and other sampling errors.

However, it's important to remember that simply increasing the sample size does not guarantee accurate estimates. If your data collection methods are flawed or your sample is not representative of the population, even a large sample size will not overcome these limitations.

Maximum Likelihood Estimation (MLE) in Detail

The Maximum Likelihood Estimator (MLE) is a powerful and widely used method for estimating parameters. The basic idea behind MLE is to find the parameter value that maximizes the likelihood function. The likelihood function represents the probability of observing the given data under different values of the parameter.

Steps in MLE:

Define the Likelihood Function: This function depends on the assumed probability distribution of the data. For example, if you assume that the data follows a normal distribution, the likelihood function will involve the normal probability density function. The likelihood function is a function of the parameters, given the data.
Maximize the Likelihood Function: This is typically done using calculus. We take the derivative of the likelihood function with respect to the parameter(s), set it equal to zero, and solve for the parameter(s). In practice, it is often easier to maximize the log-likelihood function instead of the likelihood function itself. The logarithm is a monotonic transformation, so maximizing the log-likelihood is equivalent to maximizing the likelihood.
Check Second-Order Conditions: To ensure that you have found a maximum (and not a minimum or a saddle point), you need to check the second-order conditions. This involves calculating the second derivative of the log-likelihood function and verifying that it is negative at the estimated parameter value.

Advantages of MLE:

Consistency: Under fairly general conditions, MLEs are consistent.
Asymptotic Efficiency: MLEs are asymptotically efficient, meaning that they achieve the lowest possible variance as the sample size approaches infinity.
Invariance Property: If θ̂ is the MLE of θ, then g(θ̂) is the MLE of g(θ), where g is any function. This property is useful for estimating functions of parameters.

Disadvantages of MLE:

Computational Complexity: Maximizing the likelihood function can be computationally challenging, especially for complex models.
Sensitivity to Model Assumptions: MLEs are sensitive to the assumptions made about the probability distribution of the data. If the model is misspecified, the MLEs may be biased and inconsistent.
Small Sample Size Issues: MLEs may not perform well with small sample sizes.

Examples of Estimators in Different Fields

The concept of estimators is applied across various fields:

Economics: Economists use estimators to model economic relationships, forecast economic variables (e.g., GDP, inflation), and evaluate the impact of government policies. Examples include estimating the parameters of a demand curve or a production function.
Finance: Financial analysts use estimators to assess investment risks, price financial assets, and manage portfolios. Examples include estimating the expected return and volatility of a stock.
Engineering: Engineers use estimators for signal processing, control systems, and system identification. Examples include estimating the parameters of a filter or a control law.
Medicine: Medical researchers use estimators to analyze clinical trial data, assess the effectiveness of treatments, and predict disease outcomes. Examples include estimating the survival rate of patients with a particular disease.
Social Sciences: Social scientists use estimators to study human behavior, analyze social trends, and evaluate the impact of social programs. Examples include estimating the effect of education on earnings or the impact of crime prevention programs on crime rates.

Choosing the Right Estimator

Selecting the appropriate estimator for a given situation depends on several factors, including:

The parameter being estimated: Different estimators are designed to estimate different parameters (e.g., mean, variance, proportion).
The properties of the estimator: Consider the bias, variance, consistency, and efficiency of different estimators.
The sample size: Some estimators perform better with small sample sizes, while others require large sample sizes to achieve good performance.
The assumptions about the data: Some estimators are more robust to violations of assumptions than others.
The computational complexity: Some estimators are easier to compute than others.

In practice, there is often a trade-off between different properties. For example, an unbiased estimator may have a higher variance than a biased estimator. The best estimator to use will depend on the specific goals of the analysis and the relative importance of different properties. Often, comparing several estimators and evaluating their performance using simulations or real data is useful to make the most informed decision.

Conclusion

Understanding the concept of an estimator is vital for anyone working with data and making statistical inferences. Recognizing what constitutes an estimator – a rule or function applied to sample data to estimate a population parameter – is the first step. Evaluating estimators based on their properties like unbiasedness, efficiency, consistency, and sufficiency helps determine their suitability for specific tasks. Furthermore, being aware of common estimators and their applications across diverse fields empowers individuals to make informed decisions based on statistical evidence. While selecting the right estimator involves considering various factors, the ultimate goal is to obtain reliable and accurate estimates that contribute to a deeper understanding of the population under study.