Efficient Estimator - Relative Efficiency

Relative Efficiency

If and are estimators for the parameter, then is said to dominate if:

  1. its mean squared error (MSE) is smaller for at least some value of
  2. the MSE does not exceed that of for any value of θ.

Formally, dominates if

\mathrm{E} \left[ (T_1 - \theta)^2 \right] \leq \mathrm{E} \left[ (T_2-\theta)^2 \right]

holds for all, with strict inequality holding somewhere.

The relative efficiency is defined as

e(T_1,T_2) = \frac {\mathrm{E} \left} {\mathrm{E} \left}

Although is in general a function of, in many cases the dependence drops out; if this is so, being greater than one would indicate that is preferable, whatever the true value of .

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