Master the theoretical foundations of statistical estimation, from basic concepts to advanced efficiency theory and optimal estimator construction
What you'll master in point estimation theory and applications
Master fundamental concepts of point estimation theory and evaluation criteria
Understand Method of Moments, Maximum Likelihood Estimation, and Least Squares methods
Learn Uniformly Minimum Variance Unbiased Estimators (UMVUE) and construction methods
Apply Cramér-Rao inequality and Fisher information in estimation efficiency analysis
Analyze estimator properties: unbiasedness, efficiency, consistency, and asymptotic normality
Solve practical estimation problems in statistical inference and data analysis
Three fundamental approaches to parameter estimation
Population k-th moment: μₖ = E[Xᵏ]
Sample k-th moment: aₙ,ₖ = (1/n)∑Xᵢᵏ
Parameter equation: θⱼ = hⱼ(μ₁,...,μₖ)
Moment estimator: θ̂ⱼ = hⱼ(aₙ,₁,...,aₙ,ₖ)
Example 1: Exponential E(λ) - Population moment μ₁ = 1/λ, so λ̂ = 1/X̄
Example 2: Uniform U(a,b) - μ₁ = (a+b)/2, ν₂ = (b-a)²/12, so â = X̄ - √3Sₙ
Example 3: Binomial B(k,p) - μ₁ = kp, ν₂ = kp(1-p), so p̂ = (X̄ - Sₙ²)/X̄
Example 4: Normal N(μ,σ²) - μ̂ = X̄ (sample mean), σ̂² = Sₙ² (sample variance)
Likelihood function: L(θ;x) = ∏p(xᵢ;θ)
Log-likelihood: ℓ(θ;x) = log L(θ;x)
Likelihood equation: ∂ℓ/∂θᵢ = 0
MLE: θ̂ = arg max L(θ;x)
Example 1: Normal N(μ,σ²) - L(μ,σ²) = ∏(1/(σ√2π))exp(-(xᵢ-μ)²/(2σ²)), giving μ̂ = X̄, σ̂² = Sₙ²
Example 2: Exponential E(λ) - L(λ) = λⁿexp(-λ∑xᵢ), giving λ̂ = 1/X̄
Example 3: Poisson P(λ) - L(λ) = ∏(λˣⁱe^(-λ)/xᵢ!), giving λ̂ = X̄
Example 4: Uniform U(0,θ) - L(θ) = θ^(-n) (non-differentiable), giving θ̂ = X₍ₙ₎
Model: Yᵢ = μᵢ(θ) + εᵢ
Objective: Q(θ) = ∑(Yᵢ - μᵢ(θ))²
LSE: θ̂ = arg min Q(θ)
Normal equations: ∂Q/∂θᵢ = 0
Example 1: Simple linear regression Yᵢ = β₀ + β₁xᵢ + εᵢ - β̂₁ = ∑(xᵢ-x̄)(Yᵢ-Ȳ)/∑(xᵢ-x̄)²
Example 2: Polynomial fitting Y = β₀ + β₁x + β₂x² + ε - minimize ∑(Yᵢ - β₀ - β₁xᵢ - β₂xᵢ²)²
Example 3: Exponential model Y = αe^(βx) + ε - linearize via log transform and apply LSE
Example 4: Multiple regression Y = Xβ + ε - β̂ = (X'X)⁻¹X'Y (matrix form)
How to assess and compare the quality of statistical estimators
E_θ[θ̂] = θ for all θ ∈ Θ
Why it matters: Ensures estimator does not systematically over or underestimate the parameter
Var_θ[θ̂₁] ≤ Var_θ[θ̂₂] for all θ
Why it matters: Lower variance implies more precise estimation with smaller confidence intervals
θ̂ₙ →^P θ (weak consistency)
Why it matters: Guarantees estimator accuracy improves with larger samples
MSE_θ[θ̂] = Var_θ[θ̂] + Bias²_θ[θ̂]
Why it matters: Comprehensive measure balancing accuracy and precision
Fisher information and fundamental bounds on estimator variance
I(θ) = E_θ[(∂log p(X;θ)/∂θ)²]
Var_θ[ĝ] ≥ [g'(θ)]²/Iₙ(θ)
Var_θ[ĝ*] ≤ Var_θ[ĝ] for all unbiased ĝ
Further developments in estimation theory
Deepen your understanding with practice problems, formulas, and applications
Test your understanding with carefully designed problems covering all estimation methods
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