Sample means follow a normal distribution as n increases, regardless of population shape.
X̄ ∼ N(μ, σ²/n)
An estimator is unbiased if the expected value equals the true parameter.
E(θ̂) = θ
Example: E(X̄) = μ
Bootstrap estimates variability by resampling with replacement from the original data.
Bootstrap SE Formula:
SE_B(θ̂) = √[ (1 / (n_B - 1)) Σ (θ̂_Bᵢ - θ̄_B)² ]
Used to estimate uncertainty for any statistic: mean, SD, median, regression, etc.
MLE chooses parameter values that maximize the likelihood of observed data.
L(p) = p^k · (1 - p)^(n - k)
MLE: p̂ = k / n
Assuming normal errors, MLE estimates match those from Ordinary Least Squares (OLS).
β₁̂ = Σ(Xᵢ - X̄)(Yᵢ - Ȳ) / Σ(Xᵢ - X̄)²
β₀̂ = Ȳ - β₁̂ X̄
ŷ = β₀̂ + β₁̂ X