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

Let X1,...,Xn be a random sample from a distribution with mean μ and finite variance σ². Which of the following is a sufficient pair of conditions for an estimator θ̂n of θ to be (weakly) consistent?

Bias and variance both tending to zero as n grows is sufficient for consistency, because the mean square error, the sum of variance and squared bias, then tends to zero, which forces convergence in probability to the true parameter value.

  1. ABias of θ̂n tends to 0 and variance of θ̂n tends to 0 as n tends to infinityCorrect
  2. BBias of θ̂n equals 0 for every n
  3. CVariance of θ̂n is smaller than that of every other estimator for each n
  4. Dθ̂n is a function of a sufficient statistic
  5. The mean square error of θ̂n is constant in n

Explanation

MSE = variance + bias². If bias and variance both tend to 0, MSE tends to 0, which implies convergence in probability to θ by Chebyshev/Markov. Unbiasedness alone does not give consistency, and minimum variance or sufficiency do not by themselves ensure convergence.

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