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An unbiased estimator is a statistical technique used to estimate a population parameter, where the expected value of the estimator equals the true parameter value, ensuring that it does not systematically overestimate or underestimate. This property is crucial for ensuring the accuracy and reliability of statistical inferences drawn from sample data.
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The Cramér-Rao lower bound provides a theoretical lower limit on the variance of unbiased estimators of a parameter, indicating the best precision that can be achieved by any estimator given the data. It is a fundamental result in estimation theory, enabling the assessment of estimator efficiency and guiding the development of optimal estimation techniques.
Maximum Likelihood Estimation (MLE) is a statistical method for estimating the parameters of a model by maximizing the likelihood function, thereby making the observed data most probable under the assumed statistical model. It is widely used due to its desirable properties such as consistency, efficiency, and asymptotic normality, which make it a cornerstone of statistical inference and machine learning.
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Asymptotic efficiency is a property of an estimator in statistics that describes how well the estimator performs relative to the best possible estimator as the sample size approaches infinity. It is crucial in comparing the long-term performance of different estimators, especially when sample sizes are large, to ensure that the most accurate and reliable estimator is used.
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Fisher Information measures the amount of information that an observable random variable carries about an unknown parameter upon which the probability depends. It plays a crucial role in statistical estimation, influencing the precision of parameter estimates and the design of experiments.
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Bayesian Estimation is a statistical method that updates the probability estimate for a hypothesis as more evidence or information becomes available, using Bayes' Theorem as its foundation. It provides a flexible framework for incorporating prior knowledge and observed data to make probabilistic inferences about unknown parameters.
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Estimator consistency refers to the property of an estimator to converge in probability to the true parameter value as the sample size increases. It is a fundamental criterion for evaluating the performance of an estimator in statistical inference, ensuring that with enough data, the estimator will yield accurate results.
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Estimation variance refers to the variability of an estimator's sampling distribution, reflecting how much the estimator would vary if different samples were taken from the same population. Minimizing Estimation variance is crucial for achieving more reliable and precise parameter estimates in statistical analysis.
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A sufficient statistic is a function of the data that encapsulates all necessary information needed to compute any estimate of a parameter, making other data redundant for parameter estimation. It helps in simplifying complex data by reducing it to a necessary and sufficient form, thus facilitating efficient statistical analysis.
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Robust estimation refers to statistical techniques designed to provide reliable parameter estimates even when the data contains outliers or deviates from standard assumptions, such as normality. These methods enhance the resilience of statistical analysis by reducing the influence of anomalous data points, ensuring more accurate and generalizable results.
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Generalized Least Squares (GLS) is an extension of Ordinary Least Squares (OLS) that accounts for heteroscedasticity or correlation in the error terms, providing more efficient and unbiased parameter estimates when these assumptions are violated. By transforming the model or using a weighted approach, GLS minimizes the sum of squared residuals in a way that accounts for the variance structure of the errors, leading to more reliable statistical inferences.
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