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

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 method of moments, what is the estimate of θ, and what is the estimate of the standard deviation of the distribution?

The method of moments gives θ = 6 (thousand rupees) since the exponential mean equals θ and the sample mean is 24/4 = 6. The exponential standard deviation also equals θ, so it is 6. The figure 36 is the variance.

  1. Aθ = 6, standard deviation = 36
  2. Bθ = 6, standard deviation = 6Correct
  3. Cθ = 24, standard deviation = 24
  4. Dθ = 6, standard deviation = 2.45
  5. θ = 3, standard deviation = 3

Explanation

The mean is E[X] = θ, so θ-hat = sample mean = (2+5+8+9)/4 = 24/4 = 6. For the exponential, variance = θ², so standard deviation = θ = 6. The value 36 is the variance, not the standard deviation, and 2.45 is the square root of 6.

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