IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
A random sample X1, X2, ..., X10 is taken from a distribution with mean μ and variance σ². Which of the following estimators of μ is unbiased and has the smallest variance?
The sample mean of all ten observations is correct. It is unbiased for μ and has variance σ²/10, smaller than the variance of any other unbiased option. The other estimators are either biased or use fewer observations, giving larger variance.
- A(X1 + X10)/2, with variance σ²/2
- BThe sample mean of all ten observations, with variance σ²/10Correct
- C0.9 times the sample mean, with variance 0.81σ²/10
- DX1, with variance σ²
- The sample mean plus 1/10, with variance σ²/10
Explanation
The sample mean has E = μ and variance σ²/10. (X1+X10)/2 is unbiased but has variance σ²/2, which is larger. 0.9 times the mean has bias -0.1μ, and adding 1/10 gives bias 0.1. X1 is unbiased but has variance σ². So only the sample mean is both unbiased and of minimum variance.
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