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IAI Actuarial Core Principles · Actuarial Statistics · Estimators and their properties

X1,...,Xn is a random sample from a uniform distribution on (0, θ). Let M = max(Xi). Which statement is correct?

The sample maximum is biased downward for every n, since its mean is nθ/(n+1), but both its bias and its variance tend to zero, so it is consistent for θ.

  1. AM is unbiased for θ and consistent
  2. BM is biased for every n, and not consistent
  3. CM is biased for every n but consistent, since E[M] = nθ/(n+1) and the variance tends to 0Correct
  4. DM is unbiased but not consistent
  5. M is consistent only if θ is known

Explanation

E[M]=nθ/(n+1), so the bias is -θ/(n+1), which tends to 0. Var(M)=nθ²/((n+1)²(n+2)), which tends to 0. So M is biased but consistent.

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