4.8.1 Properties of Point Estimator
Introduction¶
- Overview: Discusses the essential properties that make certain sample statistics effective point estimators for population parameters, such as mean (μ), standard deviation (σ), and proportion (p).
Desirable Properties of Point Estimators¶
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Unbiasedness: An estimator is unbiased if its expected value equals the population parameter it estimates. This means, on average, it does not overestimate or underestimate the parameter.
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Efficiency: Among unbiased estimators, an efficient estimator has the smallest variance. It provides estimates closer to the population parameter more consistently than other estimators with larger variances.
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Consistency: An estimator is consistent if it converges to the actual value of the population parameter as the sample size increases. Larger samples provide a more accurate estimation.
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Sufficiency: A sufficient estimator uses all available data and provides the most comprehensive estimate of a population parameter. It encompasses all information present in the data about the parameter.
Application and Examples¶
- Sample Mean (x̄) and Sample Proportion (p̄):
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Both x̄ and p̄ are shown to be unbiased, efficient, consistent, and sufficient, using all data points in their calculation.
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Sample Variance (s²):
- It is unbiased because it divides by n-1, allowing it to account for the degrees of freedom consumed by estimating the mean from the data.
Practical Implications¶
- Estimator Selection: The properties of point estimators influence which statistics are used in practice. For instance, the mean is preferred over the median in many cases due to its efficiency and consistency, despite the median's robustness to outliers.
Conclusion¶
- Importance of Properties: The core properties discussed ensure that the statistical methods used in inference provide reliable, accurate, and interpretable results, guiding the choice of estimators in research and application scenarios.
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