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Semester 3: Estimation Theory
Point estimation: Consistency, Unbiasedness, Efficiency, Estimators based on sufficient statistics
Point Estimation
Consistency
Consistency refers to the property of an estimator to converge in probability to the true parameter value as the sample size increases. An estimator is said to be consistent if, as the sample size approaches infinity, the probability that the estimator deviates from the true value by more than any given amount approaches zero.
Unbiasedness
An estimator is considered unbiased if the expected value of the estimator equals the true parameter value. This means that on average, the estimator hits the target parameter across multiple samples. It does not imply that any single estimate will be close to the parameter value.
Efficiency
Efficiency relates to how much information an estimator captures from the data. An efficient estimator has the smallest variance among all unbiased estimators. The Cramer-Rao Lower Bound provides a benchmark for the lowest variance possible for an unbiased estimator.
Sufficient Statistics
Sufficient statistics summarize all the information in the sample relevant for estimating the parameter. An estimator based on sufficient statistics will have the same variance as the best unbiased estimator possible, making it a powerful tool in point estimation.
References to Context
In the context of estimation theory within a B.Sc. Statistics curriculum, understanding these properties allows students to evaluate the quality of various estimators. Mastery of these concepts is crucial for advanced statistical analysis.
Minimum variance unbiased estimators, Cramer-Rao Inequality, Rao-Blackwell theorem
Estimation Theory
Minimum Variance Unbiased Estimator
The Minimum Variance Unbiased Estimator MVUE is a statistic that estimates a parameter with the least variance among all unbiased estimators. The general properties of MVUE include being unbiased and having the smallest variance for all unbiased estimators. The method to find an MVUE involves the use of the Lehmann-Scheffé theorem, which states that if a statistic is unbiased and is a function of a complete sufficient statistic, then it is the MVUE.
Cramer-Rao Inequality
The Cramer-Rao Inequality provides a lower bound for the variance of unbiased estimators. Specifically, for an unbiased estimator θ-hat of a parameter θ, the inequality states that the variance of the estimator is greater than or equal to the inverse of the Fisher Information I(θ). It expresses the efficiency of an estimator: the closer the variance of an estimator is to the bound, the more efficient the estimator is. The concept is fundamental in establishing the limitations on the precision of estimation.
Rao-Blackwell Theorem
The Rao-Blackwell theorem is a method for improving an unbiased estimator. It states that if θ-hat is an unbiased estimator of θ and T is a sufficient statistic for θ, then the conditional expectation E(θ-hat | T) is also an unbiased estimator of θ and has variance less than or equal to that of θ-hat. This theorem emphasizes the importance of sufficiency in constructing more efficient estimators and is widely used in statistical inference.
Methods of Estimation: Maximum likelihood, Method of moments
Introduction to Estimation
Estimation is a statistical process used to infer the value of an unknown parameter based on observed data. It plays a crucial role in statistics as it provides tools to summarize data and indicate population characteristics.
Maximum Likelihood Estimation (MLE)
Maximum Likelihood Estimation is a method that estimates parameters by maximizing the likelihood function. The likelihood function measures the probability of observed data under different parameter values. MLE provides estimates that have desirable properties such as consistency and efficiency.
Properties of MLE
MLE is known for its asymptotic properties. As sample size increases, the MLE estimates converge to the true parameter values, and the distribution of the estimates approaches normality. MLE is also asymptotically unbiased and achieves the Cramér-Rao lower bound.
Method of Moments
The Method of Moments is a technique for estimating parameters by equating sample moments to population moments. The first moment is the mean, the second is the variance, and so on. This method is simpler than MLE and works well for many distributions.
Comparison of MLE and Method of Moments
While MLE often provides more efficient estimators than the method of moments, the latter can be easier to compute, especially in simple cases. Both methods have their own applications and limitations depending on the context of the data and the parameter of interest.
Applications of Estimation Methods
Estimation methods are widely used in various fields, including economics, biology, engineering, and social sciences. They are fundamental in areas such as hypothesis testing, predictive modeling, and data analysis, allowing researchers to make informed decisions based on observed data.
Method of Minimum Chi-Square, Method of Minimum Variance, Method of Least squares, Interval estimation
Estimation Theory
Method of Minimum Chi-Square
The method of minimum chi-square is used in statistical estimation, particularly in the context of fitting a statistical model to observed data. It minimizes the chi-square statistic, which measures the discrepancy between observed and expected frequencies. This method is especially useful for categorical data and does not assume a normal distribution of errors.
Method of Minimum Variance
The method of minimum variance aims to estimate parameters in such a way that the variance of the estimators is minimized. This method is grounded in the Cramer-Rao lower bound which states that no unbiased estimator can have a variance lower than this bound. It is applicable in various estimation frameworks, promoting efficiency in estimating population parameters.
Method of Least Squares
The method of least squares is a widely used approach for estimating the parameters of a statistical model. It minimizes the sum of the squares of the residuals, which are the differences between observed values and those predicted by the model. This technique is foundational in regression analysis and helps in finding the best-fitting line or curve for a given set of data.
Interval Estimation
Interval estimation provides a range of values, derived from sample data, that is likely to contain the population parameter. It involves constructing confidence intervals, which quantify the uncertainty around a parameter estimate. The width of the interval is influenced by sample size and variability, and a common confidence level is 95%, indicating that if the same sampling process is repeated, approximately 95% of intervals will contain the parameter.
Bayes estimation: Prior, Posterior, Conjugate priors, Minimax estimation
Bayes estimation
Prior
The prior distribution reflects the beliefs about a parameter before observing any data. It is a key component of Bayesian inference, as it combines with the likelihood of the observed data to form the posterior distribution. Priors can be informative, reflecting knowledge based on previous studies, or non-informative, representing a state of ignorance.
Posterior
The posterior distribution is the updated belief about a parameter after observing the data. It is computed using Bayes' theorem, which relates the prior distribution, the likelihood of the observed data, and the marginal likelihood. The posterior reflects all available information about the parameter and is essential for making statistical inferences.
Conjugate priors
Conjugate priors are specific types of prior distributions that, when combined with a likelihood from a certain distribution family, result in a posterior distribution of the same family. This property simplifies the computation of the posterior. For instance, if the likelihood is binomial, a beta distribution serves as a conjugate prior.
Minimax estimation
Minimax estimation focuses on minimizing the maximum possible loss or error. In the Bayesian context, it involves choosing an estimator that minimizes the worst-case scenario under a given loss function. This approach is especially useful when dealing with uncertain parameters, providing a balance between bias and variance.
