IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
X1,...,Xn are iid Poisson(λ). Which of the following is a consistent estimator of e^(-λ) (the probability of zero)?
e raised to minus the sample mean is consistent for e^(-λ), because the sample mean converges in probability to λ and the exponential function is continuous, so the continuous mapping theorem preserves the convergence.
- Ae^(-X̄) where X̄ is the sample meanCorrect
- B1/X̄
- CX̄ e^(-X̄) only when n is even
- DThe sample variance S²
- e^(X̄)
Explanation
X̄ converges in probability to λ. By the continuous mapping theorem, e^(-X̄) converges in probability to e^(-λ). The other options converge to other quantities such as 1/λ, λ or e^λ.
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