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

Which of the following is a sufficient condition for an estimator θ̂_n of θ to be (weakly) consistent?

The estimator is consistent if both its bias and its variance tend to zero as n grows. Then the mean square error tends to zero, which implies convergence in probability to the true parameter. Either condition on its own is not enough.

  1. ABias of θ̂_n tends to 0 and variance of θ̂_n tends to 0 as n tends to infinityCorrect
  2. BBias of θ̂_n tends to 0 as n tends to infinity, with no condition on the variance
  3. CVariance of θ̂_n tends to 0 as n tends to infinity, with no condition on the bias
  4. Dθ̂_n is unbiased for every finite n
  5. θ̂_n is a function of a sufficient statistic

Explanation

If bias and variance both tend to zero then the mean square error tends to zero, which implies convergence in probability to θ. Asymptotic unbiasedness alone, or small variance alone, does not rule out the estimator settling at the wrong value. Unbiasedness for finite n (e.g. X1 for the mean) does not give consistency, and sufficiency is unrelated.

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