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

A random sample X1,...,Xn is taken from a uniform distribution on (0, θ). The estimator T = max(X1,...,Xn) has E[T] = nθ/(n+1). For a sample of n = 9 from a population with θ = 50 (rupees lakh), what is the bias of T as an estimator of θ?

The bias is -5. The expected value of the sample maximum is 9×50/10 = 45, which is 5 below the true θ of 50. The maximum always lies below θ, so it systematically underestimates it and the bias is negative.

  1. A-5Correct
  2. B-4.5
  3. C0
  4. D4.5
  5. 5

Explanation

E[T] = 9×50/10 = 45. Bias = E[T] − θ = 45 − 50 = −5. Option -4.5 wrongly uses θ/(n+1)×... with n instead of n+1 (50/11 etc. is not it); the sign positive would ignore that T underestimates θ.

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