IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
X1,...,Xn are iid with mean μ and variance σ² (both finite). Which estimator of μ is NOT consistent?
The first observation alone is not consistent, because its distribution never changes as n increases, so its variance stays at σ² and it does not converge in probability to μ. The other estimators have bias and variance tending to zero.
- AThe sample mean
- BX1 alone (the first observation)Correct
- CThe sample median of a symmetric distribution with positive density at μ
- DThe mean of the last n/2 observations
- (1/n)ΣXi + 1/√n
Explanation
X1 has variance σ² for every n, so it does not concentrate around μ and does not converge in probability. The others have variance tending to 0, and the bias in the last option is 1/√n, which tends to 0.
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