IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties
A sample of n independent observations is taken from a uniform distribution on (0, θ). The estimator θ̂ = 2X̄ is used for θ. What is its mean squared error?
The mean squared error is θ²/(3n). The estimator 2X̄ is unbiased, so MSE equals its variance, which is 4 times Var(X̄) = 4·θ²/(12n).
- Aθ²/(3n)Correct
- Bθ²/(12n)
- Cθ²/n
- Dθ²/(6n)
- 4θ²/(3n)
Explanation
E[2X̄] = θ so it is unbiased and MSE equals its variance. Var(X) = θ²/12, so Var(X̄) = θ²/(12n) and Var(2X̄) = 4θ²/(12n) = θ²/(3n). The option θ²/(12n) forgets the factor 4.
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
- Which statement about the estimator S² = (1/n)Σ(Xi − X̄)² of σ², for a random sample of size n from a normal distribution, is correct?
- X1,...,Xn are independent N(μ, σ²) with μ known to be 0. The estimator S = (1/n)ΣXi² is used for σ². Another analyst uses the estimator V = …
- For a random sample of size n from a Normal(μ, σ²) distribution with σ² = 16 known, what is the variance of the sample mean as an estimator …
- X1,...,Xn is a random sample from a normal distribution with variance σ². Estimator S² is the sample variance with divisor n-1, with varianc…
- X1,...,Xn are iid with mean μ and variance σ² (both finite). Which estimator of μ is NOT consistent?
- Let X1,...,Xn be a random sample from a distribution with mean μ and finite variance σ². Which of the following is a sufficient pair of cond…