IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
X1,...,Xn are independent Poisson(λ) variables. An analyst estimates λ² by T = (X̄)². Which expression gives the bias of T?
The bias is λ/n. Since E[X̄²] equals the variance of X̄, which is λ/n, plus the squared mean λ², the expectation exceeds λ² by λ/n. The estimator therefore overestimates λ² on average.
- A0
- Bλ/nCorrect
- C-λ/n
- Dλ²/n
- λ/n²
Explanation
E[X̄²] = Var(X̄) + (E[X̄])² = λ/n + λ². Bias = E[T] − λ² = λ/n. It is positive, so T overestimates λ².
Did you get it right without looking?
One question tells you little. A timed set on Estimators and their properties shows your real accuracy, how long you take and where you lose marks.
More Estimators and their properties questions
- X1,...,Xn are iid Poisson(λ). Which of the following is a consistent estimator of e^(-λ) (the probability of zero)?
- 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 unbiase…
- Claim sizes (in ₹ thousand) are modelled as exponential with density f(x) = (1/θ)e^(-x/θ). A sample of 4 claims is 2, 5, 8 and 9. Using the …
- X1,...,Xn is a random sample from a uniform distribution on (0, θ). Let M = max(Xi). Which statement is correct?
- For a sample of size n from a Poisson distribution with mean λ, the maximum likelihood estimate of P(X = 0) is required. With n = 50 and a s…
- A random sample of size n is taken from a Normal(μ, σ²) distribution. Consider the estimator S*² = (1/n)Σ(Xi - X̄)² of σ². Which statement i…