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Select any question to start practicing. Questions cover Fisher information, UMVUE theory, estimation methods, and efficiency concepts.

Fisher Information

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Question 1: Normal Distribution Fisher Information

For a normal distribution X ~ N(μ,σ²) with known σ², the Fisher information I(μ) for the population ...

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UMVUE Theory

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Question 2: Bernoulli UMVUE

For a sample X₁, X₂, ..., Xₙ ~ B(1,p), the UMVUE of p is:...

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Method of Moments

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Question 3: Uniform Distribution Parameter Estimation

For a uniform distribution U(a,b), the method of moments estimator for parameter a is:...

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Maximum Likelihood Estimation

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Question 4: MLE Invariance Property

For a normal distribution N(μ,σ²) with known MLE σ̂² = Sₙ² = (1/n)∑(Xᵢ-X̄)², the MLE of σ is:...

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Cramér-Rao Lower Bound

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Question 5: Poisson Distribution CR Bound

For a Poisson distribution P(λ) with sample X₁, X₂, ..., Xₙ, the Cramér-Rao lower bound for unbiased...

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Sufficient Statistics

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Question 6: Normal Distribution Sufficient Statistic

Which statistic is a sufficient complete statistic for the normal distribution N(μ,σ²)?...

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Mean Squared Error

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Question 7: MSE Decomposition

The correct decomposition of Mean Squared Error (MSE) is:...

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Least Squares Estimation

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Question 8: Linear Regression LSE

In the linear regression model Yᵢ = β₀ + β₁xᵢ + εᵢ, the least squares estimator for β₁ is:...

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Consistency

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Question 9: Uniform Distribution MLE Consistency

For uniform distribution U(0,θ) with MLE θ̂ = X(n) (maximum order statistic), the consistency proper...

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Efficiency

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Question 10: Cramér-Rao Efficiency

An estimator ĝ that achieves the Cramér-Rao lower bound is called:...

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Point Estimation & Cramér-Rao Theory Practice

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