Efficiency

**math beginner** · November 2nd 2008, 11:06 AM

Could anyone help with this?

If X1, X2, and X3 constitute a random sample of size n=3 from a normal population with the mean m and the variance s2, find the efficiency of (X1+2X2+X4)/4 relative to (X1+X2+X3)/3.

**CaptainBlack** · November 2nd 2008, 11:18 PM

Originally Posted by math beginner

Could anyone help with this?

If X1, X2, and X3 constitute a random sample of size n=3 from a normal population with the mean m and the variance s2, find the efficiency of (X1+2X2+X4)/4 relative to (X1+X2+X3)/3.

$T_1=(X_1+2X_2+X_3)/4$ and $T_2=(X_1+X_2+X_3)/3$ are unbiased estimators for \mu. (As can be seen since $E(T_1)=\mu$ and $E(T_2)=\mu$ )

Then the relative efficiency $e(T_1,T_2)$ of $T_1$ and $T_2$ is:

$e(T_1,T_2)=\frac{E((T_2-\mu)^2)}{E((T_1-\mu)^2)}$

CB

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