IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
A random sample of n = 20 is drawn from an exponential distribution with mean μ. The sample mean X̄ is unbiased for μ. Using X̄, a firm estimates the rate λ = 1/μ by 1/X̄. Given that E[1/X̄] = nλ/(n−1), what is the unbiased estimator of λ based on this result?
The unbiased estimator is 19/(20X̄). Since 1/X̄ has expectation 20λ/19, scaling it by 19/20 gives expectation exactly λ. Multiplying by 20/19 instead would worsen the overestimation rather than correct it.
- A1/X̄
- B19/(20 X̄)Correct
- C20/(19 X̄)
- D21/(20 X̄)
- 1/(19 X̄)
Explanation
E[1/X̄] = 20λ/19. To remove the bias, multiply by (n−1)/n = 19/20, giving 19/(20X̄). Check: E = (19/20)(20λ/19) = λ. Option 20/(19X̄) multiplies by the wrong factor and increases the bias.
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