Glossary

9.1 Point estimation and properties of estimators

4 min read•august 14, 2024

Point estimation is a crucial technique in inferential statistics. It uses sample data to make educated guesses about population parameters. This method helps researchers draw conclusions about large groups based on smaller, more manageable samples.

Estimators, the tools used in point estimation, have important properties like and . These qualities ensure that our estimates are reliable and improve with larger sample sizes. Understanding these concepts is key to making accurate inferences about populations.

Point Estimation in Inferential Statistics

Definition and Role of Point Estimation

Top images from around the web for Definition and Role of Point Estimation

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

1 of 3

Top images from around the web for Definition and Role of Point Estimation

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

  • Why It Matters: Linking Probability to Statistical Inference | Concepts in Statistics View original

    Is this image relevant?

1 of 3
  • Point estimation is a method in inferential statistics that uses sample data to provide a single "best guess" or estimate of a population parameter
  • The goal of point estimation is to find a statistic that is close to the unknown population parameter (population mean, proportion)
  • Point estimates are used to make inferences about the characteristics of a population based on a representative sample when collecting data from the entire population is not feasible
  • Point estimates are single values, as opposed to interval estimates which provide a range of plausible values for the parameter

Applications and Examples of Point Estimation

  • Estimating the average income of a city's residents by surveying a random sample of households
  • Determining the proportion of defective products in a manufacturing process by inspecting a sample of the produced items
  • Predicting the winner of an election based on a poll of likely voters
  • Estimating the mean weight of a certain species of fish by measuring a sample of caught fish

Properties of Good Estimators

Unbiasedness and Consistency

  • Unbiasedness is a property of an estimator where the expected value of the estimator is equal to the true value of the parameter being estimated
    • An unbiased estimator does not systematically overestimate or underestimate the parameter
    • Example: The is an unbiased estimator of the population mean
  • Consistency is a property of an estimator where the estimator converges in probability to the true value of the parameter as the sample size increases
    • A consistent estimator becomes more accurate as more data is collected
    • Example: The sample proportion is a consistent estimator of the population proportion

Efficiency and Sufficiency

  • is a property of an estimator that refers to its variance
    • An efficient estimator has the smallest possible variance among all unbiased estimators, meaning it provides the most precise estimates with the least variability
    • Example: The is an efficient estimator of the population variance when the data is normally distributed
  • Sufficiency is another desirable property, where an estimator uses all the relevant information in the sample for estimating the parameter
    • A sufficient estimator cannot be improved by using any other statistic
    • Example: The sample mean is a sufficient estimator for the population mean when the data is normally distributed

Calculating and Interpreting Point Estimates

Calculating Point Estimates for Population Parameters

  • The sample mean (xˉ\bar{x}) is a point estimate for the population mean (μ\mu)
    • It is calculated by summing all the values in the sample and dividing by the sample size: xˉ=∑i=1nxin\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}
  • The sample proportion (p^\hat{p}) is a point estimate for the population proportion (pp)
    • It is calculated by dividing the number of successes (or individuals with the characteristic of interest) by the total sample size: p^=xn\hat{p} = \frac{x}{n}
  • The sample variance (s2s^2) and standard deviation (ss) are point estimates for the population variance (σ2\sigma^2) and standard deviation (σ\sigma), respectively
    • They measure the variability of the data around the sample mean: s2=∑i=1n(xi−xˉ)2n−1s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1} and s=s2s = \sqrt{s^2}

Interpreting Point Estimates

  • Point estimates are single values that provide a best guess for the corresponding population parameter but are subject to sampling variability and may not exactly equal the true value
  • Interpreting a point estimate involves understanding the context and limitations of the estimate
    • Example: A sample mean of 75 kg for the weight of adult males in a city provides a reasonable estimate of the population mean but may not capture the true average weight of all adult males in the city due to sampling error
  • Point estimates should be accompanied by measures of uncertainty (confidence intervals) to provide a range of plausible values for the parameter

Maximum Likelihood Estimation

Concept and Application of Maximum Likelihood Estimation

  • Maximum likelihood estimation (MLE) is a method for estimating the parameters of a probability distribution by maximizing the likelihood function
  • The likelihood function represents the joint probability of observing the sample data given the parameter values
    • It is a function of the parameters, with the sample data treated as fixed
  • MLE finds the parameter values that make the observed data most likely or probable
    • The parameter values that maximize the likelihood function are called the maximum likelihood estimates
  • MLE is a versatile approach that can be applied to various probability distributions and is often used in point estimation when the underlying distribution of the data is known or assumed

Properties and Applications of Maximum Likelihood Estimators

  • Under certain regularity conditions, maximum likelihood estimators possess desirable properties
    • Consistency: As the sample size increases, the MLE converges to the true parameter value
    • Asymptotic unbiasedness: The bias of the MLE approaches zero as the sample size increases
    • Asymptotic efficiency: The MLE achieves the lowest possible variance among all consistent estimators as the sample size increases
  • MLE can be used to estimate parameters in a wide range of applications
    • Linear regression: Estimating the coefficients that minimize the sum of squared residuals
    • Logistic regression: Estimating the coefficients that maximize the likelihood of observing the binary outcomes
    • Other statistical models (Poisson regression, survival analysis)

Key Terms to Review (17)

Bayesian estimation: Bayesian estimation is a statistical method that applies Bayes' theorem to update the probability estimate for a hypothesis as more evidence or information becomes available. This approach allows for the incorporation of prior knowledge alongside observed data to produce more accurate and reliable estimates of parameters. It emphasizes the use of probability distributions to represent uncertainty in parameter values, making it a key concept in point estimation and understanding properties of estimators.
Bhattacharyya Bound: The Bhattacharyya Bound is a statistical measure that provides an upper limit on the probability of error when distinguishing between two probability distributions. It quantifies the overlap between these distributions and is often used in the context of estimation and decision-making, making it essential for evaluating the performance of estimators in point estimation problems.
Binomial Distribution: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. This distribution connects to various concepts like conditional probabilities, as it relies on the outcomes of repeated trials, and the law of large numbers, which describes how the average of results from a large number of trials tends to converge to the expected value.
Consistency: Consistency refers to a property of an estimator that indicates its reliability as the sample size increases. Specifically, a consistent estimator converges in probability to the true value of the parameter being estimated as the number of observations approaches infinity. This characteristic is crucial because it ensures that with enough data, the estimator will produce results that are closer and closer to the actual parameter value, thus providing assurance about the accuracy of the estimates derived from larger datasets.
Cramer-Rao Theorem: The Cramer-Rao Theorem states that for any unbiased estimator of a parameter, the variance of that estimator is at least as large as the inverse of the Fisher Information. This theorem provides a lower bound on the variance of estimators, helping to evaluate their efficiency and effectiveness in point estimation. It highlights the relationship between the information contained in the data and the precision of the estimators used to infer parameters.
Efficiency: Efficiency refers to the quality of an estimator in statistics that measures how well the estimator makes use of available data to produce an accurate estimate of a parameter. In this context, it connects to the precision and variability of an estimator, highlighting the balance between bias and variance. An efficient estimator has the lowest possible variance among all unbiased estimators, making it a key characteristic in evaluating point estimators.
Estimation in economics: Estimation in economics refers to the process of using statistical methods to infer the values of economic parameters or relationships based on sampled data. This involves creating models that provide a simplified representation of economic phenomena and utilizing these models to predict or estimate values that may not be directly observable. Accurate estimation is crucial for making informed decisions, policy analysis, and understanding economic trends.
Maximum Likelihood Estimator: A maximum likelihood estimator (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function, which measures how likely it is to observe the given sample data under various parameter values. MLE is particularly useful for finding point estimates that are consistent, efficient, and asymptotically normal, making it a cornerstone in point estimation and properties of estimators.
Mean Squared Error: Mean Squared Error (MSE) is a common measure used to evaluate the accuracy of an estimator by calculating the average of the squares of the errors—that is, the average squared difference between the estimated values and the actual value. This metric helps quantify how close an estimator is to the true parameter it is trying to estimate, making it essential in assessing the properties of estimators.
Method of moments: The method of moments is a technique used in statistics for estimating the parameters of a probability distribution by equating sample moments with theoretical moments. This approach allows for the estimation of population parameters such as mean and variance directly from sample data, providing a straightforward way to derive estimators.
Normal Distribution: Normal distribution is a continuous probability distribution characterized by its symmetric, bell-shaped curve, where most observations cluster around the central peak and probabilities taper off equally on both sides. This distribution is vital because many natural phenomena tend to follow this pattern, making it a foundational concept in statistics and probability.
Parameter estimation in biology: Parameter estimation in biology refers to the process of using statistical methods to estimate the values of parameters that characterize biological processes or phenomena. This is crucial for modeling and understanding various biological systems, as accurate parameter estimates help researchers make predictions, evaluate hypotheses, and gain insights into biological mechanisms.
Point Estimator: A point estimator is a statistic used to estimate the value of an unknown parameter in a population. It provides a single value as an estimate, which is derived from a sample taken from that population. The accuracy and reliability of the point estimator depend on the sampling method and the properties of the estimator itself, such as unbiasedness and consistency.
Relative Efficiency: Relative efficiency is a measure used to compare the effectiveness of two or more statistical estimators, based on their variances. It is calculated by taking the ratio of the variances of two estimators, which helps determine how much more efficient one estimator is compared to another. Understanding relative efficiency is essential for assessing the quality of different estimators in terms of their precision and reliability.
Sample mean: The sample mean is the average value of a set of data points collected from a larger population, calculated by summing all the observations and dividing by the number of observations. It serves as a point estimate of the population mean, which is crucial for understanding the overall characteristics of the population. The sample mean is foundational in statistics, especially when discussing how it behaves with larger samples and its properties as an estimator.
Sample Variance: Sample variance is a statistical measure that quantifies the degree of dispersion or variability of a set of sample data points around their mean. It is calculated by averaging the squared differences between each data point and the sample mean, giving insight into how spread out the values are in relation to the mean. This measure is critical in point estimation, helping to assess the reliability and precision of estimators used to infer population parameters.
Unbiasedness: Unbiasedness is a property of an estimator indicating that it accurately estimates a parameter without systematic errors. This means that, on average, the estimator produces values that equal the true parameter value over many samples. Unbiased estimators are crucial in statistics because they ensure that estimates are reliable and not skewed by systematic over- or underestimation.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide