📈Theoretical Statistics Unit 7 – Estimation theory

What's Estimation Theory All About?

• Estimation theory focuses on estimating unknown parameters of a population based on sample data
• Involves developing and analyzing methods to derive estimates of parameters from observed data
• Aims to minimize the difference between the estimated value and the true value of the parameter
• Deals with the properties, accuracy, and precision of different estimation techniques
• Provides a framework for quantifying uncertainty associated with estimates
• Plays a crucial role in various fields (statistics, engineering, economics, and more) where decision-making relies on estimated values

Key Concepts and Terminology

• Parameter an unknown numerical characteristic of a population (mean, variance, proportion)
• Estimator a function or rule used to estimate the value of an unknown parameter based on sample data
• Point estimate a single value that serves as the "best guess" for the unknown parameter
• Interval estimate a range of values that is likely to contain the true value of the parameter with a certain level of confidence
• Bias the difference between the expected value of an estimator and the true value of the parameter
  • An unbiased estimator has an expected value equal to the true parameter value
• Consistency an estimator is consistent if it converges to the true parameter value as the sample size increases
• Efficiency a measure of the precision of an estimator, with more efficient estimators having smaller variance

Types of Estimators

• Point estimators provide a single value as an estimate of the unknown parameter
  • Examples maximum likelihood estimator (MLE), method of moments estimator (MME)
• Interval estimators provide a range of values that likely contain the true parameter value
  • Confidence intervals are the most common type of interval estimator
• Bayesian estimators incorporate prior knowledge or beliefs about the parameter into the estimation process
  • Use Bayes' theorem to update prior beliefs with observed data to obtain a posterior distribution
• Robust estimators are less sensitive to outliers or deviations from assumptions about the underlying distribution
  • Examples median, trimmed mean, Huber estimator
• Nonparametric estimators do not rely on assumptions about the form of the underlying distribution
  • Examples kernel density estimator, empirical distribution function

Properties of Good Estimators

• Unbiasedness the expected value of the estimator should be equal to the true parameter value
• Consistency as the sample size increases, the estimator should converge to the true parameter value
• Efficiency the estimator should have the smallest possible variance among all unbiased estimators
• Sufficiency the estimator should use all relevant information contained in the sample data
• Completeness there should be a unique estimator that is unbiased and sufficient
• Minimum variance unbiased estimator (MVUE) the estimator with the smallest variance among all unbiased estimators
• Asymptotic properties desirable properties (unbiasedness, consistency, efficiency) that hold as the sample size approaches infinity

Common Estimation Methods

• Maximum likelihood estimation (MLE) chooses the parameter values that maximize the likelihood function of the observed data
  • Likelihood function $L(\theta|x)$ the probability of observing the data given the parameter values
• Method of moments estimation (MME) equates sample moments (mean, variance) to their population counterparts and solves for the parameters
• Least squares estimation (LSE) minimizes the sum of squared differences between observed and predicted values
  • Commonly used in regression analysis
• Bayesian estimation incorporates prior knowledge about the parameters and updates it with observed data using Bayes' theorem
  • Posterior distribution $p(\theta|x) \propto p(x|\theta)p(\theta)$, where $p(\theta)$ is the prior distribution and $p(x|\theta)$ is the likelihood function
• Empirical Bayes estimation uses data from related problems to estimate the prior distribution in Bayesian estimation

Practical Applications

• Estimating population characteristics (mean income, proportion of voters) from survey data
• Determining the effectiveness of a new drug or treatment in clinical trials
• Predicting future values (stock prices, weather patterns) based on historical data
• Estimating model parameters in machine learning algorithms (linear regression, logistic regression)
• Quantifying the uncertainty in engineering systems (failure rates, reliability)
• Estimating the size of wildlife populations in ecological studies
• Assessing the risk of financial investments or insurance policies

Challenges and Limitations

• Dealing with small sample sizes can lead to high uncertainty in estimates
• Violations of assumptions (normality, independence) can affect the accuracy of estimators
• Outliers or contaminated data can heavily influence some estimators
• Curse of dimensionality estimating high-dimensional parameters requires exponentially larger sample sizes
• Computational complexity some estimation methods (MLE, Bayesian) can be computationally intensive for large datasets
• Bias-variance tradeoff reducing bias often comes at the cost of increased variance, and vice versa
• Interpretability some estimators (shrinkage, regularization) may be harder to interpret than simpler methods

Advanced Topics and Future Directions

• Robust estimation developing estimators that are less sensitive to outliers or model misspecification
• High-dimensional estimation dealing with the challenges of estimating parameters in high-dimensional spaces (sparsity, regularization)
• Nonparametric estimation relaxing assumptions about the underlying distribution and using data-driven methods
• Bayesian nonparametrics combining the flexibility of nonparametric methods with the incorporati