點估計及不偏估計(Point Estimation and Unbiased Estimation)

那我們該如何衡量一個點估計的好壞呢?

Julia程式碼

using Random, Statistics
Random.seed!(0)

# 母體真實參數
trueVar, trueStd = 1/12, sqrt(1/12)

# 點估計之值
function estVar(n)
    sample = rand(n)
    sum((sample .- 0.5).^2)/n
end
# 計算點估計之期望值和母體真實參數之間是否有誤差
# 也就是說點估計是否為不偏
N = 10^7
for n in 5:5:30
    biasVar = mean([estVar(n) for _ in 1:N]) - trueVar
    biasStd = mean([sqrt(estVar(n)) for _ in 1:N]) - trueStd
    println("n = ",n, " Var bias: ", round(biasVar, digits=5),
                "\t Std bias: ", round(biasStd, digits=5))
end

Output

n = 5 Var bias: 1.0e-5

Std bias: -0.00642

n = 10 Var bias: 1.0e-5

Std bias: -0.00303

n = 15 Var bias: 0.0

Std bias: -0.00199

n = 20 Var bias: -1.0e-5

Std bias: -0.00148

n = 25 Var bias: -1.0e-5

Std bias: -0.00117

n = 30 Var bias: 0.0

Std bias: -0.00098