8.2 Properties of estimators (unbiasedness, consistency, efficiency)

Estimator properties are crucial in understanding how well statistical methods perform. Unbiasedness, consistency, and efficiency help us gauge an estimator's accuracy and precision. These concepts form the foundation for comparing different estimation techniques and selecting the most appropriate ones for specific scenarios.

Proving estimator properties involves mathematical techniques like expectation calculations and probability limit concepts. Understanding these proofs deepens our grasp of estimation theory and allows us to develop and assess new estimators. This knowledge is essential for making informed decisions in statistical analysis and research.

Estimator Properties

Unbiasedness and Consistency

Top images from around the web for Unbiasedness and Consistency

• 

  probability theory - Strong Law of Large Numbers (Klenke's proof) - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  probability - Unbiasedness of Sample Variance (missing a step in the proof) - Mathematics Stack ... View original

  Is this image relevant?

• 

  Estimating a Population Mean (1 of 3) | Concepts in Statistics View original

  Is this image relevant?

• 

  probability theory - Strong Law of Large Numbers (Klenke's proof) - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  probability - Unbiasedness of Sample Variance (missing a step in the proof) - Mathematics Stack ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Unbiasedness and Consistency

• 

  probability theory - Strong Law of Large Numbers (Klenke's proof) - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  probability - Unbiasedness of Sample Variance (missing a step in the proof) - Mathematics Stack ... View original

  Is this image relevant?

• 

  Estimating a Population Mean (1 of 3) | Concepts in Statistics View original

  Is this image relevant?

• 

  probability theory - Strong Law of Large Numbers (Klenke's proof) - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  probability - Unbiasedness of Sample Variance (missing a step in the proof) - Mathematics Stack ... View original

  Is this image relevant?

1 of 3

• Unbiasedness indicates no systematic error in estimation
  • Expected value of estimator equals true parameter value
  • Mathematically expressed as $E[\hat{\theta}] = \theta$
  • Examples: sample mean for population mean, sample proportion for population proportion
• Consistency ensures convergence to true parameter as sample size increases
  • Estimator approaches true value in probability as n approaches infinity
  • Formally written as $\lim_{n \to \infty} P(|\hat{\theta}_n - \theta| < \epsilon) = 1$ for any $\epsilon > 0$
  • Examples: sample variance for population variance, maximum likelihood estimators

Efficiency and Related Concepts

• Efficiency relates to estimator variance
  • More efficient estimators have smaller variance
  • Provide more precise parameter estimates
  • Example: sample mean is more efficient than sample median for normal distributions
• Cramér-Rao lower bound establishes minimum variance for unbiased estimators
  • Serves as efficiency benchmark
  • Expressed as $Var(\hat{\theta}) \geq \frac{1}{I(\theta)}$, where $I(\theta)$ is Fisher information
• Minimum variance unbiased estimator (MVUE) achieves Cramér-Rao lower bound
  • Most efficient among unbiased estimators
  • Example: sample mean is MVUE for population mean of normal distribution
• Bias-variance tradeoff balances accuracy and precision
  • Reducing bias often increases variance and vice versa
  • Impacts estimator selection based on specific needs
  • Example: ridge regression introduces bias to reduce variance in coefficient estimates

Estimator Quality Assessment

Comprehensive Measures

• Mean squared error (MSE) combines bias and variance
  • Calculated as $MSE = Bias^2 + Variance$
  • Lower MSE indicates better overall estimator quality
  • Example: MSE of sample mean for normal distribution is $\sigma^2/n$
• Relative efficiency compares estimator variances
  • Calculated as ratio of estimator variances
  • Values closer to 1 indicate similar efficiency
  • Example: relative efficiency of sample median to sample mean for normal distribution is $2/\pi \approx 0.637$

Desirable Properties and Considerations

• Unbiased estimators with lower variance preferred
  • Provide accurate and precise parameter estimates
  • Example: sample mean preferred over single observation for population mean
• Consistency ensures improved accuracy with larger samples
  • Crucial for large-sample inference
  • Example: law of large numbers demonstrates consistency of sample mean
• Asymptotic properties important for large sample assessment
  • Asymptotic unbiasedness
  • Asymptotic normality
  • Example: maximum likelihood estimators often possess these properties
• Robustness to assumption violations and outliers
  • Additional consideration in overall quality assessment
  • Example: median more robust than mean for skewed distributions

Estimator Comparisons

Method Comparisons

• Method of moments vs maximum likelihood estimators
  • Different finite-sample properties
  • Often asymptotically equivalent
  • Example: for exponential distribution, both yield same estimator for rate parameter
• Biased vs unbiased estimators
  • Biased estimators may outperform for small samples
  • Shrinkage estimators reduce variance at cost of bias
  • Example: James-Stein estimator outperforms sample mean in high-dimensional settings
• Consistent vs inconsistent estimators
  • Consistent estimators guarantee convergence to true value
  • Preferred for increasing sample sizes
  • Example: sample variance (consistent) vs sample range (inconsistent) for normal distribution

Selection Criteria

• Efficiency considerations for unbiased estimators
  • More efficient estimators provide smaller confidence intervals
  • Example: pooled variance estimator more efficient than separate variances for equal population variances
• Computational complexity and implementation ease
  • Balance statistical properties with practical considerations
  • Example: method of moments often simpler to compute than maximum likelihood for complex distributions
• Sample size impact on estimator choice
  • Unbiased estimators generally preferred for large samples
  • Biased estimators may be advantageous for small samples
  • Example: adjusted R-squared preferred over R-squared for small samples in multiple regression

Proving Estimator Properties

Unbiasedness Proofs

• Show expected value equals true parameter
  • Mathematically prove $E[\hat{\theta}] = \theta$
  • Example: prove unbiasedness of sample mean $E[\bar{X}] = E[\frac{1}{n}\sum_{i=1}^n X_i] = \frac{1}{n}\sum_{i=1}^n E[X_i] = \frac{1}{n}\sum_{i=1}^n \mu = \mu$
• Method of moments estimators
  • Prove sample moment expectation equals population moment
  • Example: show $E[\frac{1}{n}\sum_{i=1}^n X_i^k] = E[X^k]$ for kth moment

Consistency Proofs

• Prove convergence in probability to true parameter
  • Show $\lim_{n \to \infty} P(|\hat{\theta}_n - \theta| < \epsilon) = 1$ for any $\epsilon > 0$
• Utilize law of large numbers
  • Apply to establish consistency of sample moments
  • Example: prove consistency of sample mean using weak law of large numbers
• Employ Chebyshev's or Markov's inequality
  • Establish convergence in probability
  • Example: use Chebyshev's inequality to prove consistency of sample variance

Advanced Techniques

• Maximum likelihood estimator properties
  • Use score functions and information matrices
  • Prove asymptotic unbiasedness and efficiency
  • Example: demonstrate asymptotic normality of MLE using Taylor expansion of log-likelihood
• Invariance property of maximum likelihood estimators
  • If $\hat{\theta}$ is MLE of $\theta$, then $g(\hat{\theta})$ is MLE of $g(\theta)$
  • Example: MLE of variance is square of MLE of standard deviation
• Central limit theorem applications
  • Prove asymptotic normality of estimators
  • Example: show asymptotic normality of sample mean for non-normal populations