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

A sample of size n is drawn from a Uniform(0, θ) distribution. The estimator T = max(X1,...,Xn) has E[T] = nθ/(n+1). For n = 9, which multiple of T is an unbiased estimator of θ?

The estimator (10/9)T is unbiased. Since E[T] = 9θ/10 when n = 9, multiplying by the reciprocal 10/9 gives expectation θ. Multiplying by 9/10 or 1.1 leaves a bias, and adding θ/10 uses the unknown parameter.

  1. A0.9 T
  2. B1.1 T
  3. C(10/9) TCorrect
  4. D(9/10) T
  5. T + θ/10

Explanation

E[T] = 9θ/10 for n = 9. To get an unbiased estimator we multiply by 10/9, giving E = θ. The factor 9/10 (or 0.9) would give 0.81θ, which is biased. The factor 1.1 gives 0.99θ, also biased. T + θ/10 depends on the unknown θ and so is not an estimator.

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