An unbiased estimate has standard error 3, and the true value lies 2 from the target μ₀. You report 2/3 × x̄ + 1/3 × μ₀. What is the mean squared error of this report? Give a fraction.

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An unbiased estimate has standard error $3$ , and the true value lies $2$ from the target $μ_{0}$ . You report $\frac{2}{3} \times x ˉ + \frac{1}{3} \times μ_{0}$ . What is the mean squared error of this report? Give a fraction.

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Hint

Add the variance of the report to the square of its bias.

Solution

The variance is $\frac{2}{3} \times \frac{2}{3} \times 9 = 4$ .
The bias is $\frac{1}{3} \times 2 = \frac{2}{3}$ , and its square is $\frac{4}{9}$ .
The mean squared error is $4 + \frac{4}{9} = \frac{40}{9}$ .

Answer: $\frac{40}{9}$

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