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Semester 2: Estimation Theory
Sufficient statistics, Neyman-Fisher Factorization theorem
Sufficient Statistics and Neyman-Fisher Factorization Theorem
Definition of Sufficient Statistics
A statistic is sufficient for a parameter if it captures all the information about the parameter present in the sample data. This means that knowing the sufficient statistic gives no additional information about the parameter than knowing the data itself.
Neyman-Fisher Factorization Theorem
This theorem provides a method to identify sufficient statistics. It states that a statistic T(X) is sufficient for parameter theta if the likelihood function can be factored into two parts: one that depends only on T(X) and theta, and another that depends only on the data.
Implications of the Theorem
The Neyman-Fisher Factorization Theorem simplifies the process of finding sufficient statistics. If the likelihood is factored as described, it confirms that T(X) is a sufficient statistic.
Applications of Sufficient Statistics
Sufficient statistics are utilized in various estimation methods, particularly in Maximum Likelihood Estimation (MLE). They help to reduce the dimensionality of data, making the analysis more efficient.
Examples of Sufficient Statistics
For example, for a normally distributed population with unknown mean and known variance, the sample mean is a sufficient statistic for the mean.
Unbiased estimation, Minimum variance unbiased estimation, Rao Blackwell theorem
Estimation Theory
An unbiased estimator is a statistical estimator that correctly estimates the parameter on average. For a given parameter, an estimator is unbiased if its expected value equals the true parameter value.
E(θ̂) = θ
Unbiased estimators do not systematically overestimate or underestimate the true value.
Sample mean as an estimator of population mean.
Sample variance as an estimator of population variance.
An estimator that is both unbiased and has the lowest variance among all unbiased estimators is called a minimum variance unbiased estimator (MVUE).
MVUE achieves Cramér-Rao lower bound.
All MVUE are unbiased, but not all unbiased estimators are MVUE.
Using the sample mean for normally distributed data to estimate the population mean.
Rao-Blackwell theorem provides a method to improve any unbiased estimator.
The Rao-Blackwell theorem states that if you have an unbiased estimator and a sufficient statistic, you can obtain a better estimator by conditioning the original estimator on the sufficient statistic.
Improves estimators to achieve minimum variance.
Is widely used in the derivation of MVUE.
Deriving more efficient estimators in various statistical models.
Used in Bayesian statistics for obtaining posterior distributions.
Cramer-Rao lower bound, Bhattacharya system, Chapman-Robbins inequality
Estimation Theory
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The Cramer-Rao Lower Bound (CRLB) provides a lower bound on the variance of unbiased estimators. It states that for any unbiased estimator, the variance is at least as large as the inverse of the Fisher information.
Var(theta_hat) >= 1/I(theta) where I(theta) is the Fisher information.
CRLB is fundamental in the theory of estimation. It establishes how close an estimator can get to the best possible variance.
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The Bhattacharya system pertains to inequalities that discuss the efficiency of various estimators and their asymptotic properties. It provides a framework for analyzing the performance of estimators based on distributions.
Important in comparing estimators, particularly in the context of finite sample sizes versus asymptotic results.
Used in various statistical applications, including hypothesis testing and the evaluation of estimator properties.
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The Chapman-Robbins inequality offers a bound on the probability that an estimator will deviate from its expected value. It is useful in assessing the reliability of estimators as sample sizes increase.
P(|theta_hat - theta| >= epsilon) <= (1/epsilon^2) * Var(theta_hat) where epsilon is a small positive number.
Provides insight into the concentration of measure phenomena for estimators, which supports the understanding of convergence in probability.
Maximum likelihood estimation, Bayes and minimax estimation
Estimation Theory
Maximum Likelihood Estimation
Maximum likelihood estimation (MLE) is a method used to estimate parameters of a statistical model. MLE finds the parameter values that maximize the likelihood function, which measures how well the model explains the observed data. This technique is widely used due to its desirable properties, such as consistency and asymptotic normality. MLE requires the specification of a probability model, and calculations can be complex, often involving numerical optimization.
Bayes Estimation
Bayes estimation incorporates prior beliefs or information about parameters through the use of prior distributions. In this approach, the posterior distribution is derived using Bayes theorem, which combines the prior distribution with the likelihood of the observed data. Bayesian methods allow for the updating of beliefs as new data becomes available. This approach is particularly useful when dealing with small sample sizes or when prior information is available.
Minimax Estimation
Minimax estimation aims to minimize the maximum possible risk associated with an estimator. The estimator is chosen based on worst-case scenarios, ensuring that the worst performance across all possible parameter values is minimized. This approach is beneficial in situations where there is high uncertainty or risk, and it provides a robust framework for decision-making. Minimax procedures often lead to conservative estimators that perform well under uncertainty.
